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The cross-hatched plane is the linear span of u and v in both R 2 and R 3, here shown in perspective.. In mathematics, the linear span (also called the linear hull [1] or just span) of a set of elements of a vector space is the smallest linear subspace of that contains .
Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra , Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8 , archived from the original on March 1, 2001
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G.
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [ 1 ] [ 2 ] For example, { 3 x + 2 y − z = 1 2 x − 2 y + 4 z = − 2 − x + 1 2 y − z = 0 {\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}
In linear algebra, given a vector space with a basis of vectors indexed by an index set (the cardinality of is the dimension of ), the dual set of is a set of vectors in the dual space with the same index set such that and form a biorthogonal system.