{"id":96,"text":"equivalently a set b is a basis if its elements are linearly independent and every element of v is a linear combination of elements of b","entities":[{"id":273,"label":"Math","start_offset":52,"end_offset":72},{"id":274,"label":"Math","start_offset":101,"end_offset":119}],"relations":[{"id":137,"from_id":274,"to_id":273,"type":"Relation"}],"Comments":[]}
{"id":125,"text":"thus the solution set may be a plane a line a single point or the empty set","entities":[{"id":341,"label":"Math","start_offset":9,"end_offset":21},{"id":342,"label":"Math","start_offset":31,"end_offset":36},{"id":343,"label":"Math","start_offset":39,"end_offset":43},{"id":344,"label":"Math","start_offset":46,"end_offset":58},{"id":345,"label":"Math","start_offset":66,"end_offset":75}],"relations":[{"id":161,"from_id":345,"to_id":341,"type":"Relation"},{"id":162,"from_id":344,"to_id":341,"type":"Relation"},{"id":163,"from_id":343,"to_id":341,"type":"Relation"},{"id":164,"from_id":342,"to_id":341,"type":"Relation"}],"Comments":[]}
{"id":166,"text":"these applications may be divided into several wide categories","entities":[],"relations":[],"Comments":[]}
{"id":248,"text":"the mathematical representation of a physical vector depends on the coordinate system used to describe it","entities":[{"id":618,"label":"Math","start_offset":68,"end_offset":85},{"id":619,"label":"Math","start_offset":46,"end_offset":52}],"relations":[{"id":286,"from_id":619,"to_id":618,"type":"Relation"}],"Comments":[]}
{"id":134,"text":"in general a system with more equations than unknowns has no solution","entities":[{"id":364,"label":"Math","start_offset":30,"end_offset":39},{"id":365,"label":"Math","start_offset":61,"end_offset":69}],"relations":[{"id":170,"from_id":364,"to_id":365,"type":"Relation"}],"Comments":[]}
{"id":55,"text":"it was initially a subbranch of linear algebra but soon grew to include subjects related to graph theory algebra combinatorics and statistics","entities":[{"id":145,"label":"Math","start_offset":32,"end_offset":46},{"id":146,"label":"Math","start_offset":92,"end_offset":104},{"id":147,"label":"Math","start_offset":131,"end_offset":141},{"id":149,"label":"Math","start_offset":105,"end_offset":112},{"id":150,"label":"Math","start_offset":113,"end_offset":126}],"relations":[{"id":73,"from_id":146,"to_id":149,"type":"Relation"},{"id":74,"from_id":149,"to_id":150,"type":"Relation"},{"id":75,"from_id":150,"to_id":147,"type":"Relation"},{"id":76,"from_id":146,"to_id":145,"type":"Relation"}],"Comments":[]}
{"id":124,"text":"for three variables each linear equation determines a plane in threedimensional space and the solution set is the intersection of these planes","entities":[{"id":337,"label":"Math","start_offset":25,"end_offset":40},{"id":338,"label":"Math","start_offset":54,"end_offset":59},{"id":339,"label":"Math","start_offset":94,"end_offset":102}],"relations":[{"id":159,"from_id":339,"to_id":338,"type":"Relation"},{"id":160,"from_id":337,"to_id":338,"type":"Relation"}],"Comments":[]}
{"id":53,"text":"matrices are used in most areas of mathematics and most scientific fields either directly or through their use in geometry and numerical analysis","entities":[{"id":138,"label":"Math","start_offset":0,"end_offset":8},{"id":140,"label":"Math","start_offset":35,"end_offset":46},{"id":141,"label":"Math","start_offset":127,"end_offset":145}],"relations":[{"id":68,"from_id":138,"to_id":140,"type":"Elements"},{"id":69,"from_id":138,"to_id":141,"type":"Elements"}],"Comments":[]}
{"id":168,"text":"these are vector spaces with additional structure such as hilbert spaces","entities":[{"id":451,"label":"Math","start_offset":10,"end_offset":23},{"id":452,"label":"Math","start_offset":58,"end_offset":72}],"relations":[{"id":207,"from_id":452,"to_id":451,"type":"Elements"}],"Comments":[]}
{"id":86,"text":"these are called the trivial subspaces of the vector space","entities":[{"id":241,"label":"Math","start_offset":21,"end_offset":39},{"id":242,"label":"Math","start_offset":46,"end_offset":58}],"relations":[{"id":121,"from_id":241,"to_id":242,"type":"Elements"}],"Comments":[]}
{"id":15,"text":"many vector spaces that are considered in mathematics are also endowed with other structures","entities":[{"id":38,"label":"Math","start_offset":5,"end_offset":18},{"id":39,"label":"Math","start_offset":42,"end_offset":53}],"relations":[],"Comments":[]}
{"id":249,"text":"other vectorlike objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors","entities":[{"id":620,"label":"Math","start_offset":155,"end_offset":162}],"relations":[],"Comments":[]}
{"id":235,"text":"analysis evolved from calculus which involves the elementary concepts and techniques of analysis","entities":[{"id":583,"label":"Math","start_offset":22,"end_offset":30},{"id":584,"label":"Math","start_offset":0,"end_offset":8}],"relations":[{"id":267,"from_id":583,"to_id":584,"type":"Relation"}],"Comments":[]}
{"id":222,"text":"this line of inquiry naturally leads to the idea of the dual space the vector space v consisting of linear maps f  v  f where f is the field of scalars","entities":[{"id":553,"label":"Math","start_offset":56,"end_offset":66},{"id":554,"label":"Math","start_offset":100,"end_offset":111},{"id":555,"label":"Math","start_offset":71,"end_offset":83}],"relations":[{"id":250,"from_id":553,"to_id":555,"type":"Relation"},{"id":251,"from_id":554,"to_id":555,"type":"Elements"}],"Comments":[]}
{"id":29,"text":"most of this article deals with linear combinations in the context of a vector space over a field with some generalizations given at the end of the article","entities":[{"id":76,"label":"Math","start_offset":32,"end_offset":51},{"id":77,"label":"Math","start_offset":72,"end_offset":84}],"relations":[{"id":31,"from_id":76,"to_id":77,"type":"Elements"}],"Comments":[]}
{"id":163,"text":" in classical geometry the involved vector spaces are vector spaces over the reals but the constructions may be extended to vector spaces over any field allowing considering geometry over arbitrary fields including finite fields","entities":[{"id":440,"label":"Math","start_offset":4,"end_offset":22},{"id":441,"label":"Math","start_offset":36,"end_offset":49},{"id":442,"label":"Math","start_offset":147,"end_offset":152}],"relations":[{"id":201,"from_id":442,"to_id":441,"type":"Elements"},{"id":202,"from_id":441,"to_id":440,"type":"Relation"}],"Comments":[]}
{"id":159,"text":"this is also the case of homographies and möbius transformations when considered as transformations of a projective space","entities":[{"id":430,"label":"Math","start_offset":105,"end_offset":121}],"relations":[],"Comments":[]}
{"id":140,"text":"the third system has no solutions since the three lines share no common point","entities":[{"id":382,"label":"Math","start_offset":72,"end_offset":77}],"relations":[],"Comments":[]}
{"id":154,"text":"the basic objects of geometry which are lines and planes are represented by linear equations","entities":[{"id":414,"label":"Math","start_offset":21,"end_offset":29},{"id":415,"label":"Math","start_offset":40,"end_offset":45},{"id":416,"label":"Math","start_offset":50,"end_offset":56},{"id":417,"label":"Math","start_offset":76,"end_offset":92}],"relations":[{"id":186,"from_id":415,"to_id":414,"type":"Elements"},{"id":187,"from_id":416,"to_id":414,"type":"Elements"},{"id":188,"from_id":417,"to_id":414,"type":"Relation"}],"Comments":[]}
{"id":17,"text":"this is also the case of topological vector spaces which include function spaces inner product spaces normed spaces hilbert spaces and banach spaces","entities":[{"id":44,"label":"Math","start_offset":25,"end_offset":50},{"id":45,"label":"Math","start_offset":116,"end_offset":130},{"id":46,"label":"Math","start_offset":135,"end_offset":148}],"relations":[],"Comments":[]}
{"id":89,"text":"the linear span can be characterized either as the intersection of all linear subspaces that contain s or as the smallest subspace containing s the linear span of a set of vectors is therefore a vector space itself","entities":[{"id":250,"label":"Math","start_offset":4,"end_offset":15},{"id":251,"label":"Math","start_offset":71,"end_offset":87},{"id":253,"label":"Math","start_offset":195,"end_offset":207},{"id":255,"label":"Math","start_offset":172,"end_offset":180}],"relations":[{"id":124,"from_id":250,"to_id":251,"type":"Relation"},{"id":125,"from_id":255,"to_id":253,"type":"Elements"}],"Comments":[]}
{"id":142,"text":"it is possible for a system of two equations and two unknowns to have no solution if the two lines are parallel or for a system of three equations and two unknowns to be solvable if the three lines intersect at a single point","entities":[{"id":383,"label":"Math","start_offset":103,"end_offset":111},{"id":384,"label":"Math","start_offset":73,"end_offset":81},{"id":385,"label":"Math","start_offset":220,"end_offset":225}],"relations":[{"id":177,"from_id":385,"to_id":384,"type":"Relation"},{"id":178,"from_id":383,"to_id":384,"type":"Relation"}],"Comments":[]}
{"id":26,"text":"in mathematics a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results eg","entities":[{"id":69,"label":"Math","start_offset":17,"end_offset":35},{"id":70,"label":"Math","start_offset":42,"end_offset":52}],"relations":[{"id":27,"from_id":69,"to_id":70,"type":"Relation"}],"Comments":[]}
{"id":4,"text":"real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars real coordinate space or complex coordinate space","entities":[{"id":9,"label":"Math","start_offset":56,"end_offset":70},{"id":10,"label":"Math","start_offset":106,"end_offset":128},{"id":11,"label":"Math","start_offset":131,"end_offset":155}],"relations":[{"id":10,"from_id":9,"to_id":10,"type":"Relation"},{"id":11,"from_id":9,"to_id":11,"type":"Relation"}],"Comments":[]}
{"id":157,"text":"most geometric transformation such as translations rotations reflections rigid motions isometries and projections transform lines into lines","entities":[{"id":422,"label":"Math","start_offset":5,"end_offset":29},{"id":423,"label":"Attributes","start_offset":38,"end_offset":50},{"id":424,"label":"Attributes","start_offset":51,"end_offset":60},{"id":425,"label":"Attributes","start_offset":61,"end_offset":72},{"id":426,"label":"Attributes","start_offset":73,"end_offset":86},{"id":427,"label":"Attributes","start_offset":87,"end_offset":97},{"id":428,"label":"Attributes","start_offset":102,"end_offset":123}],"relations":[{"id":191,"from_id":423,"to_id":422,"type":"Elements"},{"id":192,"from_id":424,"to_id":422,"type":"Elements"},{"id":193,"from_id":425,"to_id":422,"type":"Elements"},{"id":194,"from_id":426,"to_id":422,"type":"Elements"},{"id":195,"from_id":427,"to_id":422,"type":"Elements"},{"id":196,"from_id":428,"to_id":422,"type":"Elements"}],"Comments":[]}
{"id":77,"text":"moreover it maps linear subspaces in  onto linear subspaces in  possibly of a lower dimension for example it maps a plane through the origin in  to either a plane through the origin in  a line through the origin in  or just the origin in","entities":[{"id":213,"label":"Math","start_offset":17,"end_offset":33},{"id":214,"label":"Math","start_offset":84,"end_offset":93}],"relations":[{"id":104,"from_id":214,"to_id":213,"type":"Elements"}],"Comments":[]}
{"id":143,"text":"a system of linear equations behave differently from the general case if the equations are linearly dependent or if it is inconsistent and has no more equations than unknowns","entities":[{"id":386,"label":"Math","start_offset":12,"end_offset":28},{"id":387,"label":"Math","start_offset":91,"end_offset":109}],"relations":[{"id":179,"from_id":387,"to_id":386,"type":"Relation"}],"Comments":[]}
{"id":21,"text":"the binary operation called vector addition or simply addition assigns to any two vectors v and w in v a third vector in v which is commonly written as v  w and called the sum of these two vectors","entities":[{"id":57,"label":"Math","start_offset":82,"end_offset":89}],"relations":[],"Comments":[]}
{"id":172,"text":"blas and lapack are the best known implementations","entities":[],"relations":[],"Comments":[]}
{"id":58,"text":"this point of view turned out to be particularly useful for the study of differential and integral equations","entities":[{"id":163,"label":"Math","start_offset":73,"end_offset":108}],"relations":[],"Comments":[]}
{"id":31,"text":"if such a linear combination exists then the vectors are said to be linearly dependent","entities":[{"id":82,"label":"Math","start_offset":10,"end_offset":28},{"id":83,"label":"Math","start_offset":68,"end_offset":86}],"relations":[{"id":34,"from_id":83,"to_id":82,"type":"Elements"}],"Comments":[]}
{"id":178,"text":"this is the case with mechanics and robotics for describing rigid body dynamics geodesy for describing earth shape perspectivity computer vision and computer graphics for describing the relationship between a scene and its plane representation and many other scientific domains","entities":[{"id":465,"label":"Attributes","start_offset":60,"end_offset":70}],"relations":[],"Comments":[]}
{"id":246,"text":" many other physical quantities can be usefully thought of as vectors","entities":[{"id":617,"label":"Math","start_offset":62,"end_offset":69}],"relations":[],"Comments":[]}
{"id":82,"text":"all these questions can be solved by using gaussian elimination or some variant of this algorithm","entities":[{"id":227,"label":"Math","start_offset":43,"end_offset":63}],"relations":[],"Comments":[]}
{"id":181,"text":"most physical phenomena are modeled by partial differential equations","entities":[{"id":470,"label":"Math","start_offset":39,"end_offset":69}],"relations":[],"Comments":[]}
{"id":22,"text":"the binary function called scalar multiplicationassigns to any scalar a in f and any vector v in v another vector in v which is denoted av","entities":[{"id":59,"label":"Math","start_offset":34,"end_offset":55},{"id":60,"label":"Math","start_offset":63,"end_offset":69},{"id":62,"label":"Math","start_offset":85,"end_offset":91}],"relations":[{"id":24,"from_id":62,"to_id":60,"type":"Relation"}],"Comments":[]}
{"id":12,"text":"finitedimensional vector spaces occur naturally in geometry and related areas","entities":[{"id":28,"label":"Math","start_offset":18,"end_offset":31},{"id":29,"label":"Math","start_offset":51,"end_offset":59},{"id":32,"label":"Attributes","start_offset":0,"end_offset":17}],"relations":[{"id":22,"from_id":32,"to_id":29,"type":"Relation"}],"Comments":[]}
{"id":195,"text":"these equations often complex and nonlinear can be linearized using linear algebra methods allowing for simpler solutions and analyses","entities":[{"id":502,"label":"Math","start_offset":68,"end_offset":82}],"relations":[],"Comments":[]}
{"id":61,"text":"however the general concept of a functional had previously been introduced in  by the italian mathematician and physicist vito volterra","entities":[{"id":166,"label":"Math","start_offset":33,"end_offset":43}],"relations":[],"Comments":[]}
{"id":9,"text":"this means that for two vector spaces over a given field and with the same dimension the properties that depend only on the vectorspace structure are exactly the same technically the vector spaces are isomorphic","entities":[{"id":22,"label":"Math","start_offset":24,"end_offset":37},{"id":23,"label":"Attributes","start_offset":201,"end_offset":211},{"id":24,"label":"Math","start_offset":75,"end_offset":85}],"relations":[{"id":18,"from_id":22,"to_id":23,"type":"Elements"},{"id":19,"from_id":22,"to_id":24,"type":"Elements"}],"Comments":[]}
{"id":5,"text":"vector spaces generalize euclidean vectors which allow modeling of physical quantities such as forces and velocity that have not only a magnitude but also a direction","entities":[{"id":12,"label":"Math","start_offset":0,"end_offset":13},{"id":13,"label":"Attributes","start_offset":136,"end_offset":145},{"id":14,"label":"Attributes","start_offset":157,"end_offset":166},{"id":15,"label":"Math","start_offset":35,"end_offset":42}],"relations":[{"id":14,"from_id":15,"to_id":13,"type":"Elements"},{"id":15,"from_id":15,"to_id":14,"type":"Elements"}],"Comments":[]}
{"id":205,"text":"it provides engineers with the necessary tools to model analyze and solve complex problems in these domains leading to advancements in technology and industry","entities":[],"relations":[],"Comments":[]}
{"id":120,"text":"this is important because if we have m independent vectors a solution is guaranteed regardless of the righthand side and otherwise not guaranteed","entities":[{"id":333,"label":"Math","start_offset":39,"end_offset":58}],"relations":[],"Comments":[]}
{"id":243,"text":"many algebraic operations on real numbers such as addition subtraction multiplication and negation have close analogues for vectors operations which obey the familiar algebraic laws of commutativity associativity and distributivity","entities":[{"id":603,"label":"Math","start_offset":5,"end_offset":25},{"id":605,"label":"Attributes","start_offset":59,"end_offset":70},{"id":606,"label":"Attributes","start_offset":50,"end_offset":58},{"id":607,"label":"Attributes","start_offset":71,"end_offset":85},{"id":608,"label":"Attributes","start_offset":90,"end_offset":98},{"id":609,"label":"Attributes","start_offset":185,"end_offset":198},{"id":610,"label":"Attributes","start_offset":199,"end_offset":212},{"id":611,"label":"Attributes","start_offset":217,"end_offset":231}],"relations":[{"id":277,"from_id":609,"to_id":603,"type":"Elements"},{"id":278,"from_id":606,"to_id":603,"type":"Elements"},{"id":279,"from_id":605,"to_id":603,"type":"Elements"},{"id":280,"from_id":607,"to_id":603,"type":"Elements"},{"id":281,"from_id":608,"to_id":603,"type":"Elements"},{"id":282,"from_id":610,"to_id":603,"type":"Elements"},{"id":283,"from_id":611,"to_id":603,"type":"Elements"}],"Comments":[]}
{"id":242,"text":" the magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from a to b","entities":[{"id":600,"label":"Math","start_offset":22,"end_offset":28},{"id":601,"label":"Attributes","start_offset":36,"end_offset":44},{"id":602,"label":"Attributes","start_offset":76,"end_offset":85}],"relations":[{"id":275,"from_id":601,"to_id":600,"type":"Elements"},{"id":276,"from_id":602,"to_id":600,"type":"Elements"}],"Comments":[]}
{"id":40,"text":"in mathematics a matrix pl","entities":[{"id":103,"label":"Math","start_offset":3,"end_offset":14},{"id":104,"label":"Math","start_offset":17,"end_offset":23}],"relations":[{"id":48,"from_id":104,"to_id":103,"type":"Elements"}],"Comments":[]}
{"id":6,"text":"the concept of vector spaces is fundamental for linear algebra together with the concept of matrices which allows computing in vector spaces","entities":[{"id":16,"label":"Math","start_offset":15,"end_offset":28},{"id":17,"label":"Math","start_offset":48,"end_offset":62},{"id":18,"label":"Math","start_offset":92,"end_offset":100}],"relations":[{"id":16,"from_id":16,"to_id":18,"type":"Elements"}],"Comments":[]}
{"id":116,"text":"this allows all the language and theory of vector spaces or more generally modules to be brought to bear","entities":[{"id":327,"label":"Math","start_offset":43,"end_offset":56}],"relations":[],"Comments":[]}
{"id":151,"text":"the solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term","entities":[{"id":404,"label":"Math","start_offset":21,"end_offset":58},{"id":406,"label":"Math","start_offset":173,"end_offset":186},{"id":407,"label":"Math","start_offset":127,"end_offset":147}],"relations":[{"id":182,"from_id":406,"to_id":407,"type":"Relation"},{"id":183,"from_id":407,"to_id":404,"type":"Relation"}],"Comments":[]}
{"id":111,"text":"gröbner basis theory provides algorithms when coefficients and unknowns are polynomials","entities":[{"id":309,"label":"Math","start_offset":30,"end_offset":40}],"relations":[],"Comments":[]}
{"id":229,"text":"a complete metric space along with the additional structure of an inner product a conjugate symmetric sesquilinear form is known as a hilbert space which is in some sense a particularly wellbehaved banach space","entities":[{"id":570,"label":"Math","start_offset":2,"end_offset":23},{"id":571,"label":"Math","start_offset":134,"end_offset":147}],"relations":[{"id":259,"from_id":570,"to_id":571,"type":"Another_name"}],"Comments":[]}
{"id":35,"text":"linear algebra is central to almost all areas of mathematics","entities":[{"id":91,"label":"Math","start_offset":0,"end_offset":14},{"id":92,"label":"Math","start_offset":49,"end_offset":60}],"relations":[{"id":38,"from_id":91,"to_id":92,"type":"Elements"}],"Comments":[]}
{"id":34,"text":"the definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space","entities":[{"id":89,"label":"Math","start_offset":18,"end_offset":35},{"id":90,"label":"Math","start_offset":98,"end_offset":110}],"relations":[{"id":37,"from_id":89,"to_id":90,"type":"Elements"}],"Comments":[]}
{"id":2,"text":"scalars are often real numbers but can be complex numbers or more generally elements of any field","entities":[{"id":7,"label":"Math","start_offset":0,"end_offset":7},{"id":8,"label":"Math","start_offset":92,"end_offset":97}],"relations":[{"id":9,"from_id":8,"to_id":7,"type":"Elements"}],"Comments":[]}
{"id":144,"text":"a differential equation can be homogeneous in either of two respects","entities":[{"id":390,"label":"Math","start_offset":2,"end_offset":23},{"id":391,"label":"Math","start_offset":31,"end_offset":42}],"relations":[{"id":180,"from_id":391,"to_id":390,"type":"Elements"}],"Comments":[]}
{"id":78,"text":"linear maps can often be represented as matrices and simple examples include rotation and reflection linear transformations","entities":[{"id":215,"label":"Math","start_offset":0,"end_offset":11},{"id":216,"label":"Math","start_offset":40,"end_offset":48},{"id":217,"label":"Attributes","start_offset":77,"end_offset":85},{"id":218,"label":"Math","start_offset":101,"end_offset":123}],"relations":[{"id":105,"from_id":218,"to_id":216,"type":"Relation"},{"id":106,"from_id":216,"to_id":215,"type":"Elements"},{"id":107,"from_id":217,"to_id":216,"type":"Relation"}],"Comments":[]}
{"id":148,"text":"which is easy to solve by integration of the two members","entities":[],"relations":[],"Comments":[]}
{"id":164,"text":"presently most textbooks introduce geometric spaces from linear algebra and geometry is often presented at elementary level as a subfield of linear algebra","entities":[{"id":443,"label":"Math","start_offset":57,"end_offset":71},{"id":444,"label":"Math","start_offset":35,"end_offset":51},{"id":445,"label":"Math","start_offset":76,"end_offset":84},{"id":446,"label":"Math","start_offset":141,"end_offset":155}],"relations":[{"id":203,"from_id":443,"to_id":444,"type":"Relation"},{"id":204,"from_id":446,"to_id":445,"type":"Relation"}],"Comments":[]}
{"id":133,"text":"in general a system with the same number of equations and unknowns has a single unique solution","entities":[{"id":362,"label":"Math","start_offset":44,"end_offset":53},{"id":363,"label":"Math","start_offset":87,"end_offset":95}],"relations":[{"id":169,"from_id":362,"to_id":363,"type":"Relation"}],"Comments":[]}
{"id":161,"text":"around this date it appeared that one may also define geometric spaces by constructions involving vector spaces see for example projective space and affine space","entities":[{"id":434,"label":"Math","start_offset":54,"end_offset":70},{"id":435,"label":"Math","start_offset":98,"end_offset":111},{"id":436,"label":"Math","start_offset":128,"end_offset":144},{"id":437,"label":"Math","start_offset":149,"end_offset":161}],"relations":[{"id":198,"from_id":436,"to_id":435,"type":"Elements"},{"id":199,"from_id":437,"to_id":435,"type":"Elements"},{"id":200,"from_id":435,"to_id":434,"type":"Relation"}],"Comments":[]}
{"id":62,"text":" the theory of nonlinear functionals was continued by students of hadamard in particular fréchet and lévy","entities":[{"id":167,"label":"Math","start_offset":1,"end_offset":37}],"relations":[],"Comments":[]}
{"id":47,"text":" this article focuses on matrices related to linear algebra and unless otherwise specified all matrices represent linear maps or may be viewed as such","entities":[{"id":119,"label":"Math","start_offset":25,"end_offset":33},{"id":120,"label":"Math","start_offset":45,"end_offset":59},{"id":121,"label":"Math","start_offset":95,"end_offset":103},{"id":122,"label":"Math","start_offset":114,"end_offset":125}],"relations":[{"id":57,"from_id":119,"to_id":120,"type":"Elements"},{"id":58,"from_id":121,"to_id":122,"type":"Elements"}],"Comments":[]}
{"id":206,"text":"the existence of multiplicative inverses in fields is not involved in the axioms defining a vector space","entities":[{"id":520,"label":"Math","start_offset":92,"end_offset":104}],"relations":[],"Comments":[]}
{"id":41,"text":" matrices is a rectangular array or table of numbers symbols or expressions arranged in rows and columns which is used to represent a mathematical object or a property of such an object","entities":[{"id":105,"label":"Math","start_offset":1,"end_offset":9}],"relations":[],"Comments":[]}
{"id":179,"text":"in all these applications synthetic geometry is often used for general descriptions and a qualitative approach but for the study of explicit situations one must compute with coordinates","entities":[{"id":466,"label":"Math","start_offset":26,"end_offset":44},{"id":468,"label":"Math","start_offset":174,"end_offset":185}],"relations":[{"id":212,"from_id":468,"to_id":466,"type":"Elements"}],"Comments":[]}
{"id":79,"text":"a bijective linear map between two vector spaces that is every vector from the second space is associated with exactly one in the first is an isomorphism","entities":[{"id":219,"label":"Math","start_offset":142,"end_offset":153}],"relations":[],"Comments":[]}
{"id":83,"text":"in mathematics and more specifically in linear algebra a linear subspace or vector subspacenote  is a vector space that is a subset of some larger vector space","entities":[{"id":230,"label":"Math","start_offset":3,"end_offset":14},{"id":231,"label":"Math","start_offset":40,"end_offset":54},{"id":232,"label":"Math","start_offset":57,"end_offset":72},{"id":233,"label":"Math","start_offset":76,"end_offset":95},{"id":234,"label":"Math","start_offset":147,"end_offset":159}],"relations":[{"id":113,"from_id":233,"to_id":232,"type":"Another_name"},{"id":114,"from_id":232,"to_id":231,"type":"Elements"},{"id":115,"from_id":231,"to_id":230,"type":"Elements"},{"id":116,"from_id":234,"to_id":231,"type":"Elements"}],"Comments":[]}
{"id":180,"text":"this requires the heavy use of linear algebra","entities":[{"id":469,"label":"Math","start_offset":31,"end_offset":45}],"relations":[],"Comments":[]}
{"id":39,"text":"for nonlinear systems which cannot be modeled with linear algebra it is often used for dealing with firstorder approximations using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point","entities":[{"id":101,"label":"Math","start_offset":208,"end_offset":218},{"id":102,"label":"Math","start_offset":51,"end_offset":65}],"relations":[{"id":47,"from_id":101,"to_id":102,"type":"Elements"}],"Comments":[]}
{"id":228,"text":"the metric also allows for a definition of limits and completeness  a metric space that is complete is known as a banach space","entities":[{"id":568,"label":"Math","start_offset":114,"end_offset":126},{"id":569,"label":"Math","start_offset":70,"end_offset":82}],"relations":[{"id":258,"from_id":568,"to_id":569,"type":"Elements"}],"Comments":[]}
{"id":158,"text":"it follows that they can be defined specified and studied in terms of linear maps","entities":[{"id":429,"label":"Math","start_offset":70,"end_offset":81}],"relations":[],"Comments":[]}
{"id":196,"text":"in the field of fluid dynamics linear algebra finds its application in computational fluid dynamics cfd a branch that uses numerical analysis and data structures to solve and analyze problems involving fluid flows","entities":[{"id":504,"label":"Math","start_offset":31,"end_offset":45}],"relations":[],"Comments":[]}
{"id":138,"text":"the first system has infinitely many solutions namely all of the points on the blue line","entities":[{"id":374,"label":"Math","start_offset":65,"end_offset":71},{"id":375,"label":"Math","start_offset":37,"end_offset":46}],"relations":[{"id":174,"from_id":375,"to_id":374,"type":"Another_name"}],"Comments":[]}
{"id":188,"text":"weather forecasting or more specifically parametrization for atmospheric modeling is a typical example of a realworld application where the whole earth atmosphere is divided into cells of say  km of width and  km of height","entities":[{"id":482,"label":"Attributes","start_offset":73,"end_offset":81}],"relations":[],"Comments":[]}
{"id":10,"text":"a vector space is finitedimensional if its dimension is a natural number","entities":[{"id":25,"label":"Math","start_offset":2,"end_offset":14},{"id":26,"label":"Attributes","start_offset":18,"end_offset":35}],"relations":[{"id":20,"from_id":25,"to_id":26,"type":"Elements"}],"Comments":[]}
{"id":182,"text":"to solve them one usually decomposes the space in which the solutions are searched into small mutually interacting cells","entities":[{"id":471,"label":"Math","start_offset":60,"end_offset":69}],"relations":[],"Comments":[]}
{"id":126,"text":"for example as three parallel planes do not have a common point the solution set of their equations is empty the solution set of the equations of three planes intersecting at a point is single point if three planes pass through two points their equations have at least two common solutions in fact the solution set is infinite and consists in all the line passing through these points","entities":[{"id":346,"label":"Math","start_offset":302,"end_offset":314}],"relations":[],"Comments":[]}
{"id":67,"text":"in mathematics differential refers to several related notions derived from the early days of calculus put on a rigorous footing such as infinitesimal differences and the derivatives of functions","entities":[{"id":181,"label":"Math","start_offset":15,"end_offset":27},{"id":182,"label":"Attributes","start_offset":136,"end_offset":161},{"id":183,"label":"Math","start_offset":170,"end_offset":181}],"relations":[{"id":90,"from_id":182,"to_id":181,"type":"Elements"},{"id":91,"from_id":183,"to_id":181,"type":"Elements"}],"Comments":[]}
{"id":65,"text":" in contrast linear algebra deals mostly with finitedimensional spaces and does not use topology","entities":[{"id":172,"label":"Math","start_offset":13,"end_offset":27},{"id":173,"label":"Math","start_offset":46,"end_offset":70},{"id":174,"label":"Math","start_offset":88,"end_offset":96}],"relations":[{"id":86,"from_id":174,"to_id":172,"type":"Elements"},{"id":87,"from_id":173,"to_id":172,"type":"Relation"}],"Comments":[]}
{"id":46,"text":"this is in particular the case in graph theory of incidence matrices and adjacency matrices","entities":[{"id":117,"label":"Math","start_offset":34,"end_offset":47},{"id":118,"label":"Math","start_offset":60,"end_offset":68}],"relations":[{"id":56,"from_id":118,"to_id":117,"type":"Elements"}],"Comments":[]}
{"id":118,"text":"if every vector within that span has exactly one expression as a linear combination of the given lefthand vectors then any solution is unique","entities":[{"id":330,"label":"Math","start_offset":9,"end_offset":15},{"id":331,"label":"Math","start_offset":65,"end_offset":83}],"relations":[{"id":158,"from_id":330,"to_id":331,"type":"Elements"}],"Comments":[]}
{"id":32,"text":"these concepts are central to the definition of dimension","entities":[{"id":84,"label":"Math","start_offset":48,"end_offset":57}],"relations":[],"Comments":[]}
{"id":191,"text":"its application in these fields is multifaceted and indispensable for solving complex problems","entities":[],"relations":[],"Comments":[]}
{"id":92,"text":"in mathematics a set b of vectors in a vector space v is called a basis pl","entities":[{"id":263,"label":"Attributes","start_offset":66,"end_offset":71},{"id":264,"label":"Math","start_offset":39,"end_offset":51},{"id":265,"label":"Math","start_offset":26,"end_offset":33}],"relations":[{"id":129,"from_id":265,"to_id":263,"type":"Another_name"},{"id":130,"from_id":264,"to_id":265,"type":"Elements"},{"id":131,"from_id":263,"to_id":264,"type":"Elements"}],"Comments":[]}
{"id":115,"text":"with respect to general linear maps linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra which is used in many parts of mathematics including geometric transformations coordinate changes quadratic forms and many other part of mathematics","entities":[{"id":320,"label":"Math","start_offset":24,"end_offset":35},{"id":321,"label":"Math","start_offset":36,"end_offset":56},{"id":322,"label":"Math","start_offset":61,"end_offset":76},{"id":324,"label":"Math","start_offset":244,"end_offset":262},{"id":325,"label":"Math","start_offset":263,"end_offset":278},{"id":326,"label":"Math","start_offset":218,"end_offset":243}],"relations":[{"id":153,"from_id":322,"to_id":321,"type":"Elements"},{"id":154,"from_id":320,"to_id":326,"type":"Relation"},{"id":155,"from_id":320,"to_id":324,"type":"Relation"},{"id":156,"from_id":320,"to_id":325,"type":"Relation"}],"Comments":[]}
{"id":177,"text":"sciences concerned with this space use geometry widely","entities":[{"id":463,"label":"Math","start_offset":39,"end_offset":47},{"id":464,"label":"Attributes","start_offset":0,"end_offset":8}],"relations":[{"id":211,"from_id":463,"to_id":464,"type":"Relation"}],"Comments":[]}
{"id":3,"text":"the operations of vector addition and scalar multiplication must satisfy certain requirements called vector axioms","entities":[],"relations":[],"Comments":[]}
{"id":187,"text":" in both cases very large matrices are generally involved","entities":[{"id":481,"label":"Math","start_offset":26,"end_offset":34}],"relations":[],"Comments":[]}
{"id":137,"text":"the following pictures illustrate this trichotomy in the case of two variables","entities":[{"id":373,"label":"Math","start_offset":69,"end_offset":78}],"relations":[],"Comments":[]}
{"id":245,"text":"vectors play an important role in physics the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors","entities":[{"id":614,"label":"Math","start_offset":0,"end_offset":7},{"id":615,"label":"Attributes","start_offset":77,"end_offset":83},{"id":616,"label":"Attributes","start_offset":99,"end_offset":105}],"relations":[{"id":284,"from_id":616,"to_id":614,"type":"Relation"},{"id":285,"from_id":615,"to_id":614,"type":"Relation"}],"Comments":[]}
{"id":193,"text":"it assists in the modeling and simulation of fluid flow providing essential tools for the analysis of fluid dynamics problems","entities":[{"id":499,"label":"Attributes","start_offset":102,"end_offset":125}],"relations":[],"Comments":[]}
{"id":209,"text":"the modules that have a basis are the free modules and those that are spanned by a finite set are the finitely generated modules","entities":[],"relations":[],"Comments":[]}
{"id":68,"text":"the term is used in various branches of mathematics such as calculus differential geometry algebraic geometry and algebraic topology","entities":[{"id":184,"label":"Math","start_offset":60,"end_offset":90},{"id":185,"label":"Math","start_offset":91,"end_offset":109},{"id":186,"label":"Math","start_offset":114,"end_offset":132}],"relations":[{"id":92,"from_id":186,"to_id":185,"type":"Relation"},{"id":93,"from_id":185,"to_id":184,"type":"Relation"}],"Comments":[]}
{"id":27,"text":"a linear combination of x and y would be any expression of the form ax  by where a and b are constants","entities":[{"id":71,"label":"Math","start_offset":2,"end_offset":20},{"id":72,"label":"Math","start_offset":45,"end_offset":55}],"relations":[{"id":28,"from_id":71,"to_id":72,"type":"Relation"}],"Comments":[]}
{"id":91,"text":"to express that a vector space v is a linear span of a subset s one commonly uses the following phraseseither s spans v s is a spanning set of v v is spannedgenerated by s or s is a generator or generator set of v","entities":[{"id":260,"label":"Math","start_offset":38,"end_offset":49},{"id":262,"label":"Math","start_offset":18,"end_offset":30}],"relations":[{"id":128,"from_id":260,"to_id":262,"type":"Elements"}],"Comments":[]}
{"id":90,"text":"spans can be generalized to matroids and modules","entities":[{"id":256,"label":"Attributes","start_offset":27,"end_offset":36},{"id":257,"label":"Attributes","start_offset":41,"end_offset":48},{"id":258,"label":"Math","start_offset":0,"end_offset":5}],"relations":[{"id":126,"from_id":256,"to_id":258,"type":"Elements"},{"id":127,"from_id":257,"to_id":258,"type":"Elements"}],"Comments":[]}
{"id":95,"text":"the elements of a basis are called basis vectors","entities":[{"id":271,"label":"Attributes","start_offset":18,"end_offset":23},{"id":272,"label":"Math","start_offset":35,"end_offset":48}],"relations":[{"id":136,"from_id":271,"to_id":272,"type":"Another_name"}],"Comments":[]}
{"id":71,"text":"f a linear map is a bijection then it is called a linear isomorphism","entities":[{"id":195,"label":"Math","start_offset":4,"end_offset":14},{"id":196,"label":"Math","start_offset":50,"end_offset":68},{"id":197,"label":"Attributes","start_offset":20,"end_offset":29}],"relations":[{"id":97,"from_id":196,"to_id":195,"type":"Relation"},{"id":98,"from_id":197,"to_id":195,"type":"Elements"}],"Comments":[]}
{"id":1,"text":"in mathematics and physics a vector space also called a linear space is a set whose elements often called vectors may be added together and multiplied scaled by numbers called scalars","entities":[{"id":3,"label":"Math","start_offset":3,"end_offset":14},{"id":4,"label":"Math","start_offset":29,"end_offset":41},{"id":5,"label":"Math","start_offset":56,"end_offset":68},{"id":6,"label":"Math","start_offset":106,"end_offset":113}],"relations":[{"id":6,"from_id":4,"to_id":5,"type":"Another_name"},{"id":7,"from_id":3,"to_id":4,"type":"Relation"},{"id":8,"from_id":4,"to_id":6,"type":"Elements"}],"Comments":[]}
{"id":70,"text":"the same names and the same definition are also used for the more general case of modules over a ring see module homomorphism","entities":[{"id":193,"label":"Math","start_offset":106,"end_offset":125}],"relations":[],"Comments":[]}
{"id":123,"text":"because a solution to a linear system must satisfy all of the equations the solution set is the intersection of these lines and is hence either a line a single point or the empty set","entities":[{"id":336,"label":"Math","start_offset":24,"end_offset":37}],"relations":[],"Comments":[]}
{"id":173,"text":"for improving efficiency some of them configure the algorithms automatically at run time for adapting them to the specificities of the computer cache size number of available cores","entities":[{"id":458,"label":"Attributes","start_offset":52,"end_offset":62}],"relations":[],"Comments":[]}
{"id":66,"text":"an important part of functional analysis is the extension of the theories of measure integration and probability to infinite dimensional spaces also known as infinite dimensional analysis","entities":[{"id":177,"label":"Math","start_offset":77,"end_offset":96},{"id":178,"label":"Math","start_offset":101,"end_offset":112},{"id":179,"label":"Math","start_offset":21,"end_offset":40}],"relations":[{"id":88,"from_id":177,"to_id":179,"type":"Elements"},{"id":89,"from_id":178,"to_id":179,"type":"Elements"}],"Comments":[]}
{"id":231,"text":"functional analysis is of particular importance to quantum mechanics the theory of partial differential equations digital signal processing and electrical engineering","entities":[{"id":576,"label":"Math","start_offset":0,"end_offset":19},{"id":577,"label":"Math","start_offset":83,"end_offset":113}],"relations":[{"id":264,"from_id":577,"to_id":576,"type":"Relation"}],"Comments":[]}
{"id":198,"text":"for example the navierstokes equations fundamental in fluid dynamics are often solved using techniques derived from linear algebra","entities":[{"id":506,"label":"Math","start_offset":116,"end_offset":130}],"relations":[],"Comments":[]}
{"id":236,"text":"analysis may be distinguished from geometry however it can be applied to any space of mathematical objects that has a definition of nearness a topological space or specific distances between objects a metric space","entities":[{"id":585,"label":"Math","start_offset":0,"end_offset":8},{"id":586,"label":"Math","start_offset":35,"end_offset":43},{"id":587,"label":"Math","start_offset":143,"end_offset":160}],"relations":[{"id":268,"from_id":586,"to_id":585,"type":"Relation"},{"id":269,"from_id":587,"to_id":586,"type":"Elements"}],"Comments":[]}
{"id":28,"text":"the concept of linear combinations is central to linear algebra and related fields of mathematics","entities":[{"id":73,"label":"Math","start_offset":15,"end_offset":34},{"id":74,"label":"Math","start_offset":49,"end_offset":63},{"id":75,"label":"Math","start_offset":86,"end_offset":97}],"relations":[{"id":29,"from_id":73,"to_id":74,"type":"Elements"},{"id":30,"from_id":74,"to_id":75,"type":"Elements"}],"Comments":[]}
{"id":81,"text":"an essential question in linear algebra is testing whether a linear map is an isomorphism or not and if it is not an isomorphism finding its range or image and the set of elements that are mapped to the zero vector called the kernel of the map","entities":[{"id":223,"label":"Math","start_offset":25,"end_offset":39},{"id":224,"label":"Math","start_offset":61,"end_offset":71},{"id":225,"label":"Attributes","start_offset":78,"end_offset":89},{"id":226,"label":"Math","start_offset":226,"end_offset":232}],"relations":[{"id":110,"from_id":225,"to_id":226,"type":"Relation"},{"id":111,"from_id":225,"to_id":224,"type":"Elements"},{"id":112,"from_id":224,"to_id":223,"type":"Elements"}],"Comments":[]}
{"id":69,"text":"in mathematics and more specifically in linear algebra a linear map also called a linear mapping linear transformation vector space homomorphism or in some contexts linear function is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication","entities":[{"id":187,"label":"Math","start_offset":40,"end_offset":54},{"id":188,"label":"Math","start_offset":57,"end_offset":67},{"id":189,"label":"Math","start_offset":82,"end_offset":97},{"id":191,"label":"Math","start_offset":97,"end_offset":118},{"id":192,"label":"Math","start_offset":119,"end_offset":144}],"relations":[{"id":94,"from_id":189,"to_id":188,"type":"Another_name"},{"id":95,"from_id":192,"to_id":188,"type":"Another_name"},{"id":96,"from_id":191,"to_id":188,"type":"Another_name"}],"Comments":[]}
{"id":204,"text":"overall the application of linear algebra in fluid mechanics fluid dynamics and thermal energy systems is an example of the profound interconnection between mathematics and engineering","entities":[{"id":516,"label":"Math","start_offset":27,"end_offset":41},{"id":517,"label":"Attributes","start_offset":45,"end_offset":60},{"id":518,"label":"Attributes","start_offset":61,"end_offset":75}],"relations":[{"id":233,"from_id":517,"to_id":516,"type":"Relation"},{"id":234,"from_id":518,"to_id":516,"type":"Relation"}],"Comments":[]}
{"id":43,"text":"therefore the study of matrices is a large part of linear algebra and most properties and operations of abstract linear algebra can be expressed in terms of matrices","entities":[{"id":109,"label":"Math","start_offset":23,"end_offset":31},{"id":110,"label":"Math","start_offset":51,"end_offset":65},{"id":111,"label":"Math","start_offset":113,"end_offset":127},{"id":112,"label":"Math","start_offset":157,"end_offset":165}],"relations":[{"id":51,"from_id":109,"to_id":110,"type":"Elements"},{"id":52,"from_id":112,"to_id":111,"type":"Elements"}],"Comments":[]}
{"id":160,"text":"until the end of the th century geometric spaces were defined by axioms relating points lines and planes synthetic geometry","entities":[{"id":431,"label":"Math","start_offset":32,"end_offset":48},{"id":432,"label":"Math","start_offset":105,"end_offset":123}],"relations":[{"id":197,"from_id":431,"to_id":432,"type":"Relation"}],"Comments":[]}
{"id":127,"text":"for n variables each linear equation determines a hyperplane in ndimensional space","entities":[{"id":348,"label":"Math","start_offset":64,"end_offset":82},{"id":349,"label":"Math","start_offset":37,"end_offset":47},{"id":350,"label":"Math","start_offset":6,"end_offset":15}],"relations":[{"id":165,"from_id":350,"to_id":349,"type":"Relation"},{"id":166,"from_id":349,"to_id":348,"type":"Relation"}],"Comments":[]}
{"id":234,"text":"these theories are usually studied in the context of real and complex numbers and functions","entities":[{"id":582,"label":"Math","start_offset":82,"end_offset":91}],"relations":[],"Comments":[]}
{"id":167,"text":"functional analysis studies function spaces","entities":[{"id":449,"label":"Math","start_offset":0,"end_offset":19},{"id":450,"label":"Math","start_offset":28,"end_offset":43}],"relations":[{"id":206,"from_id":450,"to_id":449,"type":"Elements"}],"Comments":[]}
{"id":85,"text":"as a corollary all vector spaces are equipped with at least two possibly different linear subspaces the zero vector space consisting of the zero vector alone and the entire vector space itself","entities":[{"id":238,"label":"Math","start_offset":173,"end_offset":185},{"id":239,"label":"Math","start_offset":83,"end_offset":99},{"id":240,"label":"Math","start_offset":104,"end_offset":121}],"relations":[{"id":118,"from_id":239,"to_id":238,"type":"Relation"},{"id":119,"from_id":240,"to_id":238,"type":"Relation"},{"id":120,"from_id":239,"to_id":240,"type":"Relation"}],"Comments":[]}
{"id":139,"text":"the second system has a single unique solution namely the intersection of the two lines","entities":[{"id":378,"label":"Math","start_offset":58,"end_offset":70},{"id":379,"label":"Math","start_offset":24,"end_offset":46}],"relations":[{"id":175,"from_id":379,"to_id":378,"type":"Another_name"}],"Comments":[]}
{"id":169,"text":"linear algebra is thus a fundamental part of functional analysis and its applications which include in particular quantum mechanics wave functions and fourier analysis orthogonal basis","entities":[{"id":453,"label":"Math","start_offset":0,"end_offset":14},{"id":454,"label":"Math","start_offset":45,"end_offset":64},{"id":455,"label":"Math","start_offset":151,"end_offset":184}],"relations":[{"id":208,"from_id":454,"to_id":453,"type":"Relation"},{"id":209,"from_id":455,"to_id":454,"type":"Elements"}],"Comments":[]}
{"id":7,"text":"this provides a concise and synthetic way for manipulating and studying systems of linear equations","entities":[{"id":19,"label":"Math","start_offset":83,"end_offset":99}],"relations":[],"Comments":[]}
{"id":44,"text":"for example matrix multiplication represents the composition of linear maps","entities":[{"id":113,"label":"Math","start_offset":12,"end_offset":18},{"id":114,"label":"Math","start_offset":64,"end_offset":75}],"relations":[{"id":54,"from_id":113,"to_id":114,"type":"Elements"}],"Comments":[]}
{"id":105,"text":"linear systems are the basis and a fundamental part of linear algebra a subject used in most modern mathematics","entities":[{"id":296,"label":"Math","start_offset":55,"end_offset":69},{"id":297,"label":"Math","start_offset":100,"end_offset":111},{"id":298,"label":"Math","start_offset":0,"end_offset":14}],"relations":[{"id":146,"from_id":298,"to_id":296,"type":"Relation"},{"id":147,"from_id":296,"to_id":297,"type":"Another_name"}],"Comments":[]}
{"id":183,"text":"for linear systems this interaction involves linear functions","entities":[{"id":472,"label":"Math","start_offset":4,"end_offset":18},{"id":473,"label":"Math","start_offset":45,"end_offset":61}],"relations":[{"id":213,"from_id":473,"to_id":472,"type":"Elements"}],"Comments":[]}
{"id":72,"text":"in the case where a linear map is called a linear endomorphism","entities":[{"id":198,"label":"Math","start_offset":20,"end_offset":30},{"id":199,"label":"Math","start_offset":43,"end_offset":62}],"relations":[{"id":99,"from_id":199,"to_id":198,"type":"Relation"}],"Comments":[]}
{"id":174,"text":"some processors typically graphics processing units gpu are designed with a matrix structure for optimizing the operations of linear algebra","entities":[{"id":459,"label":"Attributes","start_offset":76,"end_offset":92},{"id":460,"label":"Math","start_offset":126,"end_offset":140}],"relations":[{"id":210,"from_id":459,"to_id":460,"type":"Relation"}],"Comments":[]}
{"id":48,"text":"square matrices matrices with the same number of rows and columns play a major role in matrix theory","entities":[{"id":123,"label":"Math","start_offset":0,"end_offset":15},{"id":124,"label":"Math","start_offset":87,"end_offset":100}],"relations":[{"id":59,"from_id":123,"to_id":124,"type":"Elements"}],"Comments":[]}
{"id":197,"text":"cfd relies heavily on linear algebra for the computation of fluid flow and heat transfer in various applications","entities":[{"id":505,"label":"Math","start_offset":22,"end_offset":36}],"relations":[],"Comments":[]}
{"id":202,"text":"linear algebraic concepts such as matrix operations and eigenvalue problems are employed to enhance the efficiency reliability and economic performance of power systems","entities":[{"id":512,"label":"Math","start_offset":34,"end_offset":40},{"id":513,"label":"Math","start_offset":56,"end_offset":66},{"id":514,"label":"Math","start_offset":0,"end_offset":16}],"relations":[{"id":231,"from_id":512,"to_id":514,"type":"Elements"},{"id":232,"from_id":513,"to_id":514,"type":"Elements"}],"Comments":[]}
{"id":8,"text":"vector spaces are characterized by their dimension which roughly speaking specifies the number of independent directions in the space","entities":[{"id":20,"label":"Math","start_offset":0,"end_offset":13},{"id":21,"label":"Attributes","start_offset":41,"end_offset":50}],"relations":[{"id":17,"from_id":20,"to_id":21,"type":"Elements"}],"Comments":[]}
{"id":51,"text":"in geometry matrices are widely used for specifying and representing geometric transformations for example rotations and coordinate changes","entities":[{"id":133,"label":"Math","start_offset":12,"end_offset":20},{"id":134,"label":"Attributes","start_offset":107,"end_offset":116},{"id":135,"label":"Attributes","start_offset":121,"end_offset":139}],"relations":[{"id":65,"from_id":134,"to_id":133,"type":"Elements"},{"id":66,"from_id":135,"to_id":133,"type":"Elements"}],"Comments":[]}
{"id":218,"text":"there are many rings for which there are algorithms for solving linear equations and systems of linear equations","entities":[{"id":545,"label":"Math","start_offset":64,"end_offset":80},{"id":546,"label":"Math","start_offset":15,"end_offset":20}],"relations":[{"id":247,"from_id":546,"to_id":545,"type":"Relation"}],"Comments":[]}
{"id":227,"text":"the norm induces a metric which measures the distance between elements and induces a topology which allows for a definition of continuous maps","entities":[{"id":566,"label":"Math","start_offset":4,"end_offset":8},{"id":567,"label":"Attributes","start_offset":45,"end_offset":53}],"relations":[{"id":256,"from_id":567,"to_id":566,"type":"Relation"}],"Comments":[]}
{"id":36,"text":"for instance linear algebra is fundamental in modern presentations of geometry including for defining basic objects such as lines planes and rotations","entities":[{"id":93,"label":"Math","start_offset":13,"end_offset":27},{"id":94,"label":"Math","start_offset":70,"end_offset":78},{"id":95,"label":"Math","start_offset":124,"end_offset":136},{"id":96,"label":"Math","start_offset":141,"end_offset":150}],"relations":[{"id":39,"from_id":93,"to_id":94,"type":"Relation"},{"id":43,"from_id":95,"to_id":94,"type":"Elements"},{"id":44,"from_id":96,"to_id":94,"type":"Elements"}],"Comments":[]}
{"id":104,"text":"the word system indicates that the equations should be considered collectively rather than individually","entities":[],"relations":[],"Comments":[]}
{"id":38,"text":"linear algebra is also used in most sciences and fields of engineering because it allows modeling many natural phenomena and computing efficiently with such models","entities":[{"id":99,"label":"Math","start_offset":0,"end_offset":14}],"relations":[],"Comments":[]}
{"id":239,"text":"a euclidean vector is frequently represented by a directed line segment or graphically as an arrow connecting an initial point a with a terminal point b and denoted by","entities":[{"id":596,"label":"Math","start_offset":12,"end_offset":18}],"relations":[],"Comments":[]}
{"id":146,"text":"where f and g are homogeneous functions of the same degree of x and y","entities":[{"id":395,"label":"Math","start_offset":18,"end_offset":39}],"relations":[],"Comments":[]}
{"id":97,"text":" in other words a basis is a linearly independent spanning set","entities":[{"id":275,"label":"Attributes","start_offset":18,"end_offset":23},{"id":276,"label":"Math","start_offset":29,"end_offset":62}],"relations":[{"id":138,"from_id":275,"to_id":276,"type":"Elements"}],"Comments":[]}
{"id":75,"text":"in the language of category theory linear maps are the morphisms of vector spaces","entities":[{"id":209,"label":"Math","start_offset":19,"end_offset":34},{"id":210,"label":"Math","start_offset":35,"end_offset":46},{"id":211,"label":"Math","start_offset":68,"end_offset":81}],"relations":[{"id":102,"from_id":211,"to_id":210,"type":"Relation"},{"id":103,"from_id":210,"to_id":209,"type":"Elements"}],"Comments":[]}
{"id":107,"text":"a system of nonlinear equations can often be approximated by a linear system see linearization a helpful technique when making a mathematical model or computer simulation of a relatively complex system","entities":[{"id":301,"label":"Math","start_offset":12,"end_offset":31},{"id":302,"label":"Math","start_offset":63,"end_offset":76}],"relations":[],"Comments":[]}
{"id":84,"text":"a linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces","entities":[{"id":235,"label":"Math","start_offset":2,"end_offset":17},{"id":237,"label":"Math","start_offset":45,"end_offset":53}],"relations":[{"id":117,"from_id":237,"to_id":235,"type":"Another_name"}],"Comments":[]}
{"id":128,"text":"the solution set is the intersection of these hyperplanes and is a flat which may have any dimension lower than n","entities":[{"id":352,"label":"Math","start_offset":4,"end_offset":16},{"id":353,"label":"Math","start_offset":91,"end_offset":100},{"id":354,"label":"Math","start_offset":46,"end_offset":58}],"relations":[{"id":167,"from_id":352,"to_id":354,"type":"Relation"},{"id":168,"from_id":354,"to_id":353,"type":"Relation"}],"Comments":[]}
{"id":156,"text":"this was one of the main motivations for developing linear algebra","entities":[{"id":421,"label":"Math","start_offset":52,"end_offset":66}],"relations":[],"Comments":[]}
{"id":199,"text":"this includes the use of matrices and vectors to represent and manipulate fluid flow fields","entities":[{"id":507,"label":"Math","start_offset":25,"end_offset":33},{"id":508,"label":"Math","start_offset":38,"end_offset":45},{"id":509,"label":"Attributes","start_offset":73,"end_offset":91}],"relations":[{"id":227,"from_id":508,"to_id":507,"type":"Relation"},{"id":228,"from_id":509,"to_id":508,"type":"Relation"},{"id":229,"from_id":509,"to_id":507,"type":"Relation"}],"Comments":[]}
{"id":16,"text":"this is the case of algebras which include field extensions polynomial rings associative algebras and lie algebras","entities":[{"id":41,"label":"Math","start_offset":49,"end_offset":76}],"relations":[],"Comments":[]}
{"id":212,"text":"vector spaces are completely characterized by their dimension up to an isomorphism","entities":[{"id":530,"label":"Math","start_offset":0,"end_offset":13},{"id":531,"label":"Math","start_offset":52,"end_offset":61},{"id":532,"label":"Math","start_offset":71,"end_offset":82}],"relations":[{"id":239,"from_id":531,"to_id":530,"type":"Elements"},{"id":240,"from_id":532,"to_id":530,"type":"Elements"}],"Comments":[]}
{"id":60,"text":"the term was first used in hadamards  book on that subject","entities":[],"relations":[],"Comments":[]}
{"id":98,"text":"a vector space can have several bases however all the bases have the same number of elements called the dimension of the vector space","entities":[{"id":277,"label":"Math","start_offset":2,"end_offset":14},{"id":278,"label":"Attributes","start_offset":32,"end_offset":37},{"id":280,"label":"Math","start_offset":104,"end_offset":113}],"relations":[{"id":139,"from_id":280,"to_id":278,"type":"Relation"},{"id":141,"from_id":278,"to_id":277,"type":"Elements"}],"Comments":[]}
{"id":122,"text":"for a system involving two variables x and y each linear equation determines a line on the xyplane","entities":[{"id":335,"label":"Math","start_offset":50,"end_offset":65}],"relations":[],"Comments":[]}
{"id":112,"text":"tropical geometry is another example of linear algebra in a more exotic structure","entities":[{"id":311,"label":"Math","start_offset":0,"end_offset":17},{"id":312,"label":"Math","start_offset":39,"end_offset":54}],"relations":[{"id":149,"from_id":311,"to_id":312,"type":"Relation"}],"Comments":[]}
{"id":52,"text":"in numerical analysis many computational problems are solved by reducing them to a matrix computation and this often involves computing with matrices of huge dimension","entities":[{"id":136,"label":"Math","start_offset":3,"end_offset":21},{"id":137,"label":"Math","start_offset":83,"end_offset":90}],"relations":[{"id":67,"from_id":137,"to_id":136,"type":"Elements"}],"Comments":[]}
{"id":129,"text":"in general the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns","entities":[{"id":356,"label":"Math","start_offset":29,"end_offset":42}],"relations":[],"Comments":[]}
{"id":121,"text":"geometric interpretation","entities":[{"id":334,"label":"Math","start_offset":0,"end_offset":9}],"relations":[],"Comments":[]}
{"id":145,"text":"a first order differential equation is said to be homogeneous if it may be written","entities":[{"id":392,"label":"Math","start_offset":0,"end_offset":35},{"id":393,"label":"Math","start_offset":50,"end_offset":61}],"relations":[{"id":181,"from_id":392,"to_id":393,"type":"Another_name"}],"Comments":[]}
{"id":201,"text":"it is used to model and optimize the generation transmission and distribution of electric power","entities":[],"relations":[],"Comments":[]}
{"id":101,"text":"in mathematics a system of linear equations or linear system is a collection of one or more linear equations involving the same variables","entities":[{"id":283,"label":"Math","start_offset":3,"end_offset":14},{"id":284,"label":"Math","start_offset":27,"end_offset":43},{"id":285,"label":"Math","start_offset":47,"end_offset":60}],"relations":[{"id":142,"from_id":284,"to_id":283,"type":"Elements"},{"id":143,"from_id":285,"to_id":284,"type":"Relation"}],"Comments":[]}
{"id":109,"text":"for solutions in an integral domain like the ring of the integers or in other algebraic structures other theories have been developed see linear equation over a ring","entities":[{"id":307,"label":"Math","start_offset":138,"end_offset":153}],"relations":[],"Comments":[]}
{"id":215,"text":"modules over the integers can be identified with abelian groups since the multiplication by an integer may be identified to a repeated addition","entities":[{"id":535,"label":"Math","start_offset":0,"end_offset":7},{"id":536,"label":"Math","start_offset":49,"end_offset":63},{"id":538,"label":"Attributes","start_offset":74,"end_offset":88},{"id":539,"label":"Attributes","start_offset":135,"end_offset":143},{"id":540,"label":"Math","start_offset":17,"end_offset":25}],"relations":[{"id":241,"from_id":535,"to_id":540,"type":"Relation"},{"id":242,"from_id":536,"to_id":540,"type":"Relation"},{"id":243,"from_id":539,"to_id":535,"type":"Relation"},{"id":244,"from_id":538,"to_id":535,"type":"Relation"}],"Comments":[]}
{"id":155,"text":"thus computing intersections of lines and planes amounts to solving systems of linear equations","entities":[{"id":418,"label":"Math","start_offset":79,"end_offset":95},{"id":419,"label":"Math","start_offset":32,"end_offset":37},{"id":420,"label":"Math","start_offset":42,"end_offset":48}],"relations":[{"id":189,"from_id":419,"to_id":418,"type":"Relation"},{"id":190,"from_id":420,"to_id":418,"type":"Relation"}],"Comments":[]}
{"id":208,"text":"the concepts of linear independence span basis and linear maps also called module homomorphisms are defined for modules exactly as for vector spaces with the essential difference that if r is not a field there are modules that do not have any basis","entities":[{"id":522,"label":"Math","start_offset":16,"end_offset":46},{"id":523,"label":"Math","start_offset":51,"end_offset":62},{"id":524,"label":"Math","start_offset":75,"end_offset":95}],"relations":[{"id":235,"from_id":524,"to_id":523,"type":"Another_name"},{"id":236,"from_id":523,"to_id":522,"type":"Relation"}],"Comments":[]}
{"id":119,"text":"in any event the span has a basis of linearly independent vectors that do guarantee exactly one expression and the number of vectors in that basis its dimension cannot be larger than m or n but it can be smaller","entities":[{"id":332,"label":"Math","start_offset":125,"end_offset":132}],"relations":[],"Comments":[]}
{"id":171,"text":" consequently linear algebra algorithms have been highly optimized","entities":[{"id":457,"label":"Math","start_offset":14,"end_offset":28}],"relations":[],"Comments":[]}
{"id":244,"text":"these operations and associated laws qualify euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space","entities":[{"id":613,"label":"Math","start_offset":152,"end_offset":164}],"relations":[],"Comments":[]}
{"id":76,"text":"a linear map from  to  always maps the origin of  to the origin of","entities":[{"id":212,"label":"Math","start_offset":2,"end_offset":12}],"relations":[],"Comments":[]}
{"id":64,"text":"in modern introductory texts on functional analysis the subject is seen as the study of vector spaces endowed with a topology in particular infinitedimensional spaces","entities":[{"id":169,"label":"Math","start_offset":32,"end_offset":51},{"id":170,"label":"Math","start_offset":88,"end_offset":101},{"id":171,"label":"Math","start_offset":117,"end_offset":125}],"relations":[{"id":84,"from_id":171,"to_id":170,"type":"Relation"},{"id":85,"from_id":170,"to_id":169,"type":"Relation"}],"Comments":[]}
{"id":99,"text":"this article deals mainly with finitedimensional vector spaces","entities":[{"id":281,"label":"Math","start_offset":49,"end_offset":62}],"relations":[],"Comments":[]}
{"id":54,"text":"matrix theory is the branch of mathematics that focuses on the study of matrices","entities":[{"id":142,"label":"Math","start_offset":0,"end_offset":13},{"id":143,"label":"Math","start_offset":31,"end_offset":42},{"id":144,"label":"Math","start_offset":72,"end_offset":80}],"relations":[{"id":71,"from_id":142,"to_id":143,"type":"Elements"},{"id":72,"from_id":144,"to_id":142,"type":"Elements"}],"Comments":[]}
{"id":102,"text":"a solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied","entities":[{"id":287,"label":"Math","start_offset":2,"end_offset":10},{"id":288,"label":"Math","start_offset":64,"end_offset":74},{"id":289,"label":"Math","start_offset":106,"end_offset":120}],"relations":[{"id":144,"from_id":288,"to_id":287,"type":"Another_name"},{"id":145,"from_id":289,"to_id":288,"type":"Relation"}],"Comments":[]}
{"id":221,"text":"in multilinear algebra one considers multivariable linear transformations that is mappings that are linear in each of a number of different variables","entities":[{"id":550,"label":"Math","start_offset":3,"end_offset":22},{"id":552,"label":"Math","start_offset":51,"end_offset":73}],"relations":[{"id":249,"from_id":552,"to_id":550,"type":"Relation"}],"Comments":[]}
{"id":11,"text":"otherwise it is infinitedimensional and its dimension is an infinite cardinal","entities":[{"id":27,"label":"Attributes","start_offset":16,"end_offset":35}],"relations":[],"Comments":[]}
{"id":226,"text":"a normed vector space is a vector space along with a function called a norm which measures the size of elements","entities":[{"id":564,"label":"Math","start_offset":2,"end_offset":21},{"id":565,"label":"Math","start_offset":27,"end_offset":39}],"relations":[{"id":255,"from_id":564,"to_id":565,"type":"Elements"}],"Comments":[]}
{"id":132,"text":"such a system is known as an underdetermined system","entities":[{"id":360,"label":"Math","start_offset":29,"end_offset":51}],"relations":[],"Comments":[]}
{"id":224,"text":"if in addition to vector addition and scalar multiplication there is a bilinear vector product v  v  v the vector space is called an algebra for instance associative algebras are algebras with an associate vector product like the algebra of square matrices or the algebra of polynomials","entities":[{"id":558,"label":"Math","start_offset":18,"end_offset":33},{"id":559,"label":"Math","start_offset":38,"end_offset":59},{"id":560,"label":"Math","start_offset":71,"end_offset":94},{"id":562,"label":"Math","start_offset":107,"end_offset":119}],"relations":[{"id":253,"from_id":558,"to_id":559,"type":"Relation"},{"id":254,"from_id":560,"to_id":562,"type":"Elements"}],"Comments":[]}
{"id":233,"text":"analysis is the branch of mathematics dealing with continuous functions limits and related theories such as differentiation integration measure infinite sequences series and analytic functions","entities":[{"id":579,"label":"Math","start_offset":0,"end_offset":8},{"id":580,"label":"Math","start_offset":51,"end_offset":78},{"id":581,"label":"Math","start_offset":174,"end_offset":192}],"relations":[{"id":265,"from_id":581,"to_id":579,"type":"Relation"},{"id":266,"from_id":580,"to_id":581,"type":"Relation"}],"Comments":[]}
{"id":238,"text":"vectors can be added to other vectors according to vector algebra","entities":[{"id":593,"label":"Math","start_offset":51,"end_offset":65},{"id":595,"label":"Math","start_offset":0,"end_offset":7}],"relations":[{"id":274,"from_id":595,"to_id":593,"type":"Elements"}],"Comments":[]}