{"id":192,"text":"in fluid mechanics linear algebra is integral to understanding and solving problems related to the behavior of fluids","entities":[{"id":497,"label":"Attributes","start_offset":3,"end_offset":18},{"id":498,"label":"Math","start_offset":19,"end_offset":33}],"relations":[{"id":225,"from_id":497,"to_id":498,"type":"Relation"}],"Comments":[]}
{"id":33,"text":"a vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors","entities":[{"id":85,"label":"Math","start_offset":2,"end_offset":14},{"id":86,"label":"Math","start_offset":25,"end_offset":41},{"id":87,"label":"Math","start_offset":45,"end_offset":63}],"relations":[{"id":35,"from_id":86,"to_id":85,"type":"Elements"},{"id":36,"from_id":87,"to_id":85,"type":"Elements"}],"Comments":[]}
{"id":175,"text":"citation needed","entities":[],"relations":[],"Comments":[]}
{"id":211,"text":"the theory of matrices over a ring is similar to that of matrices over a field except that determinants exist only if the ring is commutative and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring","entities":[{"id":526,"label":"Math","start_offset":14,"end_offset":22},{"id":527,"label":"Math","start_offset":30,"end_offset":34},{"id":528,"label":"Math","start_offset":73,"end_offset":78},{"id":529,"label":"Math","start_offset":57,"end_offset":65}],"relations":[{"id":237,"from_id":526,"to_id":527,"type":"Relation"},{"id":238,"from_id":529,"to_id":528,"type":"Relation"}],"Comments":[]}
{"id":73,"text":"sometimes the term linear operator refers to this case but the term linear operator can have different meanings for different conventions for example it can be used to emphasize that and are real vector spaces not necessarily with citation needed or it can be used to emphasize that  is a function space which is a common convention in functional analysis","entities":[{"id":201,"label":"Math","start_offset":19,"end_offset":34},{"id":203,"label":"Math","start_offset":335,"end_offset":355},{"id":204,"label":"Math","start_offset":289,"end_offset":303}],"relations":[],"Comments":[]}
{"id":19,"text":"a vector space over a field f is a nonempty set v together with a binary operation and a binary function that satisfy the eight axioms listed below","entities":[{"id":49,"label":"Math","start_offset":2,"end_offset":14},{"id":50,"label":"Math","start_offset":22,"end_offset":28},{"id":53,"label":"Math","start_offset":35,"end_offset":47}],"relations":[],"Comments":[]}
{"id":186,"text":"linear models are frequently used for complex nonlinear realworld systems because it makes parametrization more manageable","entities":[{"id":478,"label":"Math","start_offset":0,"end_offset":13},{"id":479,"label":"Attributes","start_offset":38,"end_offset":73}],"relations":[{"id":216,"from_id":478,"to_id":479,"type":"Relation"}],"Comments":[]}
{"id":230,"text":"functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces the central objects of study in functional analysis are lp spaces which are banach spaces and especially the l space of square integrable functions which is the only hilbert space among them","entities":[{"id":572,"label":"Math","start_offset":0,"end_offset":19},{"id":573,"label":"Math","start_offset":43,"end_offset":57},{"id":574,"label":"Math","start_offset":208,"end_offset":221},{"id":575,"label":"Math","start_offset":298,"end_offset":311}],"relations":[{"id":260,"from_id":572,"to_id":573,"type":"Relation"},{"id":261,"from_id":574,"to_id":575,"type":"Relation"},{"id":263,"from_id":574,"to_id":572,"type":"Relation"}],"Comments":[]}
{"id":147,"text":" in this case the change of variable y  ux leads to an equation of the form","entities":[{"id":398,"label":"Math","start_offset":55,"end_offset":63}],"relations":[],"Comments":[]}
{"id":14,"text":"for example polynomial rings are countably infinitedimensional vector spaces and many function spaces have the cardinality of the continuum as a dimension","entities":[{"id":35,"label":"Math","start_offset":12,"end_offset":28},{"id":36,"label":"Attributes","start_offset":43,"end_offset":62},{"id":37,"label":"Math","start_offset":63,"end_offset":76}],"relations":[],"Comments":[]}
{"id":189,"text":"fluid mechanics fluid dynamics and thermal energy systems","entities":[{"id":483,"label":"Attributes","start_offset":0,"end_offset":15},{"id":485,"label":"Attributes","start_offset":16,"end_offset":30},{"id":486,"label":"Attributes","start_offset":35,"end_offset":57}],"relations":[{"id":217,"from_id":486,"to_id":485,"type":"Relation"},{"id":218,"from_id":485,"to_id":483,"type":"Relation"},{"id":219,"from_id":483,"to_id":486,"type":"Relation"}],"Comments":[]}
{"id":190,"text":"linear algebra a branch of mathematics dealing with vector spaces and linear mappings between these spaces plays a critical role in various engineering disciplines including fluid mechanics fluid dynamics and thermal energy systems","entities":[{"id":488,"label":"Math","start_offset":0,"end_offset":14},{"id":489,"label":"Math","start_offset":27,"end_offset":38},{"id":490,"label":"Math","start_offset":52,"end_offset":65},{"id":491,"label":"Math","start_offset":70,"end_offset":85},{"id":492,"label":"Attributes","start_offset":174,"end_offset":189},{"id":493,"label":"Attributes","start_offset":190,"end_offset":204},{"id":495,"label":"Attributes","start_offset":209,"end_offset":231}],"relations":[{"id":220,"from_id":488,"to_id":489,"type":"Elements"},{"id":221,"from_id":491,"to_id":490,"type":"Elements"},{"id":222,"from_id":492,"to_id":488,"type":"Relation"},{"id":223,"from_id":493,"to_id":488,"type":"Relation"},{"id":224,"from_id":495,"to_id":488,"type":"Relation"}],"Comments":[]}
{"id":103,"text":"in the example above a solution is given by the ordered triple  since it makes all three equations valid","entities":[{"id":291,"label":"Math","start_offset":23,"end_offset":31},{"id":293,"label":"Math","start_offset":89,"end_offset":98}],"relations":[],"Comments":[]}
{"id":110,"text":"integer linear programming is a collection of methods for finding the best integer solution when there are many","entities":[{"id":308,"label":"Math","start_offset":0,"end_offset":26}],"relations":[],"Comments":[]}
{"id":108,"text":"very often and in this article the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers but the theory and the algorithms apply for coefficients and solutions in any field","entities":[{"id":304,"label":"Math","start_offset":195,"end_offset":207},{"id":305,"label":"Math","start_offset":212,"end_offset":221}],"relations":[],"Comments":[]}
{"id":114,"text":"if v has a basis of n elements such an endomorphism is represented by a square matrix of size n","entities":[{"id":318,"label":"Math","start_offset":11,"end_offset":16},{"id":319,"label":"Math","start_offset":72,"end_offset":85}],"relations":[{"id":152,"from_id":318,"to_id":319,"type":"Elements"}],"Comments":[]}
{"id":45,"text":"not all matrices are related to linear algebra","entities":[{"id":115,"label":"Math","start_offset":8,"end_offset":16},{"id":116,"label":"Math","start_offset":32,"end_offset":46}],"relations":[{"id":55,"from_id":115,"to_id":116,"type":"Elements"}],"Comments":[]}
{"id":93,"text":" bases if every element of v may be written in a unique way as a finite linear combination of elements of b","entities":[{"id":266,"label":"Attributes","start_offset":1,"end_offset":6},{"id":267,"label":"Math","start_offset":72,"end_offset":90}],"relations":[{"id":132,"from_id":266,"to_id":267,"type":"Elements"}],"Comments":[]}
{"id":149,"text":"otherwise a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives","entities":[{"id":401,"label":"Math","start_offset":12,"end_offset":33},{"id":402,"label":"Math","start_offset":60,"end_offset":80}],"relations":[],"Comments":[]}
{"id":18,"text":"in this article vectors are represented in boldface to distinguish them from scalars","entities":[{"id":47,"label":"Math","start_offset":16,"end_offset":23},{"id":48,"label":"Math","start_offset":77,"end_offset":84}],"relations":[],"Comments":[]}
{"id":131,"text":"in general a system with fewer equations than unknowns has infinitely many solutions but it may have no solution","entities":[{"id":359,"label":"Math","start_offset":31,"end_offset":40}],"relations":[],"Comments":[]}
{"id":162,"text":"it has been shown that the two approaches are essentially equivalent","entities":[{"id":439,"label":"Math","start_offset":58,"end_offset":68}],"relations":[],"Comments":[]}
{"id":219,"text":"however these algorithms have generally a computational complexity that is much higher than the similar algorithms over a field","entities":[],"relations":[],"Comments":[]}
{"id":216,"text":"most of the theory of abelian groups may be extended to modules over a principal ideal domain","entities":[{"id":541,"label":"Math","start_offset":22,"end_offset":36},{"id":542,"label":"Math","start_offset":56,"end_offset":63}],"relations":[{"id":245,"from_id":542,"to_id":541,"type":"Relation"}],"Comments":[]}
{"id":170,"text":"nearly all scientific computations involve linear algebra","entities":[{"id":456,"label":"Math","start_offset":43,"end_offset":57}],"relations":[],"Comments":[]}
{"id":176,"text":"the modeling of ambient space is based on geometry","entities":[{"id":462,"label":"Math","start_offset":42,"end_offset":50}],"relations":[],"Comments":[]}
{"id":153,"text":"in this new at that time geometry now called cartesian geometry points are represented by cartesian coordinates which are sequences of three real numbers in the case of the usual threedimensional space","entities":[{"id":412,"label":"Math","start_offset":25,"end_offset":33},{"id":413,"label":"Math","start_offset":45,"end_offset":63}],"relations":[{"id":185,"from_id":413,"to_id":412,"type":"Relation"}],"Comments":[]}
{"id":220,"text":"for more details see linear equation over a ring","entities":[{"id":547,"label":"Math","start_offset":21,"end_offset":36},{"id":548,"label":"Math","start_offset":44,"end_offset":48}],"relations":[{"id":248,"from_id":547,"to_id":548,"type":"Relation"}],"Comments":[]}
{"id":63,"text":"hadamard also founded the modern school of linear functional analysis further developed by riesz and the group of polish mathematicians around stefan banach","entities":[{"id":168,"label":"Math","start_offset":43,"end_offset":69}],"relations":[],"Comments":[]}
{"id":23,"text":"to have a vector space the eight following axioms must be satisfied for every u v and w in v and a and b in f","entities":[{"id":63,"label":"Math","start_offset":10,"end_offset":22}],"relations":[],"Comments":[]}
{"id":25,"text":"these two cases are the most common ones but vector spaces with scalars in an arbitrary field f are also commonly considered","entities":[{"id":67,"label":"Math","start_offset":45,"end_offset":58},{"id":68,"label":"Math","start_offset":64,"end_offset":71}],"relations":[],"Comments":[]}
{"id":56,"text":"functional analysis is a branch of mathematical analysis the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure for example inner product norm or topology and the linear functions defined on these spaces and suitably respecting these structures","entities":[{"id":151,"label":"Math","start_offset":0,"end_offset":19},{"id":152,"label":"Math","start_offset":35,"end_offset":56},{"id":153,"label":"Math","start_offset":101,"end_offset":114},{"id":155,"label":"Math","start_offset":176,"end_offset":189},{"id":156,"label":"Math","start_offset":190,"end_offset":194},{"id":157,"label":"Math","start_offset":198,"end_offset":206},{"id":158,"label":"Math","start_offset":215,"end_offset":231}],"relations":[{"id":77,"from_id":151,"to_id":152,"type":"Elements"},{"id":78,"from_id":155,"to_id":153,"type":"Elements"},{"id":79,"from_id":156,"to_id":153,"type":"Elements"},{"id":80,"from_id":157,"to_id":153,"type":"Elements"},{"id":81,"from_id":158,"to_id":151,"type":"Relation"}],"Comments":[]}
{"id":59,"text":"the usage of the word functional as a noun goes back to the calculus of variations implying a function whose argument is a function","entities":[{"id":164,"label":"Math","start_offset":72,"end_offset":82},{"id":165,"label":"Math","start_offset":123,"end_offset":131}],"relations":[],"Comments":[]}
{"id":87,"text":"if v is a vector space over a field k and if w is a subset of v then w is a linear subspace of v if under the operations of v w is a vector space over k equivalently a nonempty subset w is a linear subspace of v if whenever w w are elements of w and α β are elements of k it follows that αw  βw is in w","entities":[{"id":243,"label":"Math","start_offset":191,"end_offset":206},{"id":244,"label":"Math","start_offset":10,"end_offset":23}],"relations":[{"id":122,"from_id":243,"to_id":244,"type":"Elements"}],"Comments":[]}
{"id":49,"text":"square matrices of a given dimension form a noncommutative ring which is one of the most common examples of a noncommutative ring","entities":[{"id":125,"label":"Math","start_offset":0,"end_offset":15},{"id":126,"label":"Math","start_offset":27,"end_offset":36},{"id":127,"label":"Math","start_offset":44,"end_offset":63}],"relations":[{"id":60,"from_id":126,"to_id":125,"type":"Elements"},{"id":61,"from_id":125,"to_id":127,"type":"Elements"}],"Comments":[]}
{"id":194,"text":"for instance linear algebraic techniques are used to solve systems of differential equations that describe fluid motion","entities":[{"id":500,"label":"Math","start_offset":13,"end_offset":29},{"id":501,"label":"Attributes","start_offset":107,"end_offset":119}],"relations":[{"id":226,"from_id":500,"to_id":501,"type":"Relation"}],"Comments":[]}
{"id":165,"text":"linear algebra is used in almost all areas of mathematics thus making it relevant in almost all scientific domains that use mathematics","entities":[{"id":447,"label":"Math","start_offset":0,"end_offset":14},{"id":448,"label":"Math","start_offset":46,"end_offset":57}],"relations":[{"id":205,"from_id":447,"to_id":448,"type":"Elements"}],"Comments":[]}
{"id":214,"text":"however every module is a cokernel of a homomorphism of free modules","entities":[{"id":533,"label":"Math","start_offset":14,"end_offset":20}],"relations":[],"Comments":[]}