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William Gilbert Strang (born November 27, 1934 [1]) is an American mathematician known for his contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing mathematics textbooks.
Here we provide two proofs. The first [2] operates in the general case, using linear maps. The second proof [6] looks at the homogeneous system =, where is a with rank, and shows explicitly that there exists a set of linearly independent solutions that span the null space of .
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In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation .
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
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. For three variables, each linear equation determines a plane in three-dimensional space , and the solution set is the intersection of these planes.
This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices. Linear equations
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.. More precisely, if , …, are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between , …, is a sequence (, …,) of elements of R such that
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