{"id":185,"text":"bthis is called a linear model or firstorder approximation","entities":[{"id":476,"label":"Math","start_offset":18,"end_offset":30},{"id":477,"label":"Math","start_offset":34,"end_offset":58}],"relations":[{"id":215,"from_id":476,"to_id":477,"type":"Relation"}],"Comments":[]}
{"id":232,"text":"it also provides the foundation and theoretical framework that underlies the fourier transform and related methods","entities":[{"id":578,"label":"Math","start_offset":77,"end_offset":94}],"relations":[],"Comments":[]}
{"id":117,"text":"for example the collection of all possible linear combinations of the vectors on the lefthand side is called their span and the equations have a solution just when the righthand vector is within that span","entities":[{"id":328,"label":"Math","start_offset":43,"end_offset":62},{"id":329,"label":"Math","start_offset":70,"end_offset":77}],"relations":[{"id":157,"from_id":329,"to_id":328,"type":"Elements"}],"Comments":[]}
{"id":94,"text":"the coefficients of this linear combination are referred to as components or coordinates of the vector with respect to b","entities":[{"id":268,"label":"Math","start_offset":63,"end_offset":73},{"id":269,"label":"Math","start_offset":77,"end_offset":88},{"id":270,"label":"Math","start_offset":25,"end_offset":43}],"relations":[{"id":133,"from_id":268,"to_id":270,"type":"Elements"},{"id":134,"from_id":269,"to_id":268,"type":"Another_name"},{"id":135,"from_id":269,"to_id":270,"type":"Elements"}],"Comments":[]}
{"id":184,"text":"for nonlinear systems this interaction is often approximated by linear functions","entities":[{"id":474,"label":"Math","start_offset":4,"end_offset":21},{"id":475,"label":"Math","start_offset":64,"end_offset":80}],"relations":[{"id":214,"from_id":474,"to_id":475,"type":"Elements"}],"Comments":[]}
{"id":141,"text":"it must be kept in mind that the pictures above show only the most common case the general case","entities":[],"relations":[],"Comments":[]}
{"id":30,"text":"in the theory of vector spaces a set of vectors is said to be linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector","entities":[{"id":78,"label":"Math","start_offset":17,"end_offset":30},{"id":79,"label":"Math","start_offset":40,"end_offset":47},{"id":80,"label":"Math","start_offset":62,"end_offset":82},{"id":81,"label":"Math","start_offset":113,"end_offset":131}],"relations":[{"id":32,"from_id":81,"to_id":78,"type":"Elements"},{"id":33,"from_id":80,"to_id":78,"type":"Elements"}],"Comments":[]}
{"id":88,"text":"in mathematics the linear span also called the linear hull or just span of a set s of vectors from a vector space denoted spans is defined as the set of all linear combinations of the vectors in s for example two linearly independent vectors span a plane","entities":[{"id":245,"label":"Math","start_offset":3,"end_offset":15},{"id":246,"label":"Math","start_offset":19,"end_offset":30},{"id":247,"label":"Math","start_offset":47,"end_offset":58}],"relations":[{"id":123,"from_id":247,"to_id":246,"type":"Another_name"}],"Comments":[]}
{"id":100,"text":"however many of the principles are also valid for infinitedimensional vector spaces","entities":[{"id":282,"label":"Math","start_offset":70,"end_offset":83}],"relations":[],"Comments":[]}
{"id":37,"text":"also functional analysis a branch of mathematical analysis may be viewed as the application of linear algebra to function spaces","entities":[{"id":97,"label":"Math","start_offset":5,"end_offset":24},{"id":98,"label":"Math","start_offset":95,"end_offset":109}],"relations":[{"id":45,"from_id":97,"to_id":98,"type":"Relation"}],"Comments":[]}
{"id":237,"text":"in mathematics physics and engineering a euclidean vector or simply a vector sometimes called a geometric vector or spatial vector is a geometric object that has magnitude or length and direction","entities":[{"id":589,"label":"Math","start_offset":96,"end_offset":112},{"id":590,"label":"Attributes","start_offset":162,"end_offset":171},{"id":591,"label":"Attributes","start_offset":175,"end_offset":181},{"id":592,"label":"Attributes","start_offset":186,"end_offset":195}],"relations":[{"id":270,"from_id":590,"to_id":591,"type":"Elements"},{"id":271,"from_id":592,"to_id":589,"type":"Relation"},{"id":273,"from_id":591,"to_id":589,"type":"Relation"}],"Comments":[]}
{"id":241,"text":" it was first used by th century astronomers investigating planetary revolution around the sun","entities":[],"relations":[],"Comments":[]}
{"id":130,"text":"here in general means that a different behavior may occur for specific values of the coefficients of the equations","entities":[{"id":358,"label":"Math","start_offset":62,"end_offset":77}],"relations":[],"Comments":[]}
{"id":223,"text":"multilinear maps t  vn  f can be described via tensor products of elements of v","entities":[{"id":556,"label":"Math","start_offset":0,"end_offset":16},{"id":557,"label":"Math","start_offset":47,"end_offset":53}],"relations":[{"id":252,"from_id":557,"to_id":556,"type":"Elements"}],"Comments":[]}
{"id":240,"text":"a vector is what is needed to carry the point a to the point b the latin word vector means carrier","entities":[{"id":597,"label":"Math","start_offset":2,"end_offset":8}],"relations":[],"Comments":[]}
{"id":203,"text":"the application of linear algebra in this context is vital for the design and operation of modern power systems including renewable energy sources and smart grids","entities":[{"id":515,"label":"Math","start_offset":19,"end_offset":33}],"relations":[],"Comments":[]}
{"id":207,"text":"one may thus replace the field of scalars by a ring r and this gives the structure called a module over r or rmodule","entities":[{"id":521,"label":"Math","start_offset":47,"end_offset":51}],"relations":[],"Comments":[]}
{"id":24,"text":"when the scalar field is the real numbers the vector space is called a real vector space and when the scalar field is the complex numbers the vector space is called a complex vector space","entities":[{"id":64,"label":"Math","start_offset":46,"end_offset":58},{"id":65,"label":"Math","start_offset":167,"end_offset":187},{"id":66,"label":"Math","start_offset":71,"end_offset":88}],"relations":[{"id":25,"from_id":64,"to_id":66,"type":"Elements"},{"id":26,"from_id":64,"to_id":65,"type":"Elements"}],"Comments":[]}
{"id":106,"text":"computational algorithms for finding the solutions are an important part of numerical linear algebra and play a prominent role in engineering physics chemistry computer science and economics","entities":[{"id":299,"label":"Math","start_offset":86,"end_offset":100},{"id":300,"label":"Math","start_offset":41,"end_offset":50}],"relations":[],"Comments":[]}
{"id":136,"text":"in the first case the dimension of the solution set is in general equal to n  m where n is the number of variables and m is the number of equations","entities":[{"id":368,"label":"Math","start_offset":22,"end_offset":31},{"id":369,"label":"Math","start_offset":39,"end_offset":51},{"id":370,"label":"Math","start_offset":138,"end_offset":147},{"id":371,"label":"Math","start_offset":105,"end_offset":114}],"relations":[{"id":171,"from_id":369,"to_id":368,"type":"Relation"},{"id":172,"from_id":369,"to_id":371,"type":"Relation"},{"id":173,"from_id":369,"to_id":370,"type":"Relation"}],"Comments":[]}
{"id":150,"text":"in the case of linear differential equations this means that there are no constant terms","entities":[{"id":403,"label":"Math","start_offset":14,"end_offset":44}],"relations":[],"Comments":[]}
{"id":200,"text":"furthermore linear algebra plays a crucial role in thermal energy systems particularly in power systems analysis","entities":[{"id":510,"label":"Math","start_offset":12,"end_offset":26},{"id":511,"label":"Attributes","start_offset":51,"end_offset":73}],"relations":[{"id":230,"from_id":511,"to_id":510,"type":"Relation"}],"Comments":[]}
{"id":225,"text":"vector spaces that are not finite dimensional often require additional structure to be tractable","entities":[{"id":563,"label":"Math","start_offset":0,"end_offset":13}],"relations":[],"Comments":[]}
{"id":57,"text":"the historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the fourier transform as transformations defining for example continuous or unitary operators between function spaces","entities":[{"id":159,"label":"Math","start_offset":24,"end_offset":43},{"id":161,"label":"Math","start_offset":162,"end_offset":179},{"id":162,"label":"Math","start_offset":234,"end_offset":251}],"relations":[{"id":82,"from_id":161,"to_id":159,"type":"Elements"},{"id":83,"from_id":162,"to_id":159,"type":"Elements"}],"Comments":[]}
{"id":213,"text":"in general there is not such a complete classification for modules even if one restricts oneself to finitely generated modules","entities":[{"id":534,"label":"Math","start_offset":59,"end_offset":66}],"relations":[],"Comments":[]}
{"id":152,"text":"there is a strong relationship between linear algebra and geometry which started with the introduction by rené descartes in  of cartesian coordinates","entities":[{"id":410,"label":"Math","start_offset":39,"end_offset":53},{"id":411,"label":"Math","start_offset":58,"end_offset":66}],"relations":[{"id":184,"from_id":411,"to_id":410,"type":"Relation"}],"Comments":[]}
{"id":247,"text":"although most of them do not represent distances except for example position or displacement their magnitude and direction can still be represented by the length and direction of an arrow","entities":[],"relations":[],"Comments":[]}
{"id":113,"text":"a linear endomorphism is a linear map that maps a vector space v to itself","entities":[{"id":314,"label":"Math","start_offset":2,"end_offset":21},{"id":315,"label":"Math","start_offset":27,"end_offset":37},{"id":316,"label":"Math","start_offset":50,"end_offset":62}],"relations":[{"id":150,"from_id":314,"to_id":315,"type":"Relation"},{"id":151,"from_id":315,"to_id":316,"type":"Elements"}],"Comments":[]}
{"id":50,"text":"the determinant of a square matrix is a number associated to the matrix which is fundamental for the study of a square matrix for example a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant","entities":[{"id":128,"label":"Math","start_offset":4,"end_offset":15},{"id":129,"label":"Math","start_offset":21,"end_offset":34},{"id":130,"label":"Math","start_offset":220,"end_offset":231},{"id":131,"label":"Math","start_offset":237,"end_offset":250},{"id":132,"label":"Math","start_offset":270,"end_offset":292}],"relations":[{"id":62,"from_id":130,"to_id":132,"type":"Relation"},{"id":63,"from_id":130,"to_id":131,"type":"Elements"},{"id":64,"from_id":128,"to_id":129,"type":"Elements"}],"Comments":[]}
{"id":20,"text":"in this context the elements of v are commonly called vectors and the elements of f are called scalars","entities":[{"id":54,"label":"Math","start_offset":54,"end_offset":61},{"id":55,"label":"Math","start_offset":95,"end_offset":102}],"relations":[],"Comments":[]}
{"id":42,"text":"matrices are used to represent linear maps and allow explicit computations in linear algebra","entities":[{"id":106,"label":"Math","start_offset":0,"end_offset":8},{"id":107,"label":"Math","start_offset":31,"end_offset":42},{"id":108,"label":"Math","start_offset":78,"end_offset":92}],"relations":[{"id":49,"from_id":106,"to_id":107,"type":"Elements"},{"id":50,"from_id":107,"to_id":108,"type":"Elements"}],"Comments":[]}
{"id":135,"text":"such a system is also known as an overdetermined system","entities":[{"id":366,"label":"Math","start_offset":34,"end_offset":55}],"relations":[],"Comments":[]}
{"id":80,"text":"because an isomorphism preserves linear structure two isomorphic vector spaces are essentially the same from the linear algebra point of view in the sense that they cannot be distinguished by using vector space properties","entities":[{"id":220,"label":"Math","start_offset":11,"end_offset":22},{"id":221,"label":"Math","start_offset":113,"end_offset":127},{"id":222,"label":"Math","start_offset":54,"end_offset":78}],"relations":[{"id":108,"from_id":222,"to_id":221,"type":"Relation"},{"id":109,"from_id":220,"to_id":222,"type":"Relation"}],"Comments":[]}
{"id":74,"text":" sometimes the term linear function has the same meaning as linear map while in analysis it does not","entities":[{"id":205,"label":"Math","start_offset":60,"end_offset":70},{"id":208,"label":"Math","start_offset":20,"end_offset":35}],"relations":[{"id":101,"from_id":205,"to_id":208,"type":"Relation"}],"Comments":[]}
{"id":13,"text":"infinitedimensional vector spaces occur in many areas of mathematics","entities":[{"id":30,"label":"Math","start_offset":20,"end_offset":33},{"id":31,"label":"Math","start_offset":57,"end_offset":68},{"id":34,"label":"Attributes","start_offset":0,"end_offset":19}],"relations":[{"id":23,"from_id":34,"to_id":31,"type":"Relation"}],"Comments":[]}
{"id":217,"text":"in particular over a principal ideal domain every submodule of a free module is free and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring","entities":[{"id":543,"label":"Math","start_offset":50,"end_offset":59},{"id":544,"label":"Math","start_offset":70,"end_offset":76}],"relations":[{"id":246,"from_id":543,"to_id":544,"type":"Elements"}],"Comments":[]}
{"id":210,"text":"module homomorphisms between finitely generated free modules may be represented by matrices","entities":[{"id":525,"label":"Math","start_offset":83,"end_offset":91}],"relations":[],"Comments":[]}