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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 . It is the set of all finite linear combinations of the elements of S , [ 2 ] and the intersection of all linear subspaces that contain S . {\displaystyle S.}
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 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.
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .
The linear span of the -blades is called the -th exterior power of . The exterior algebra is the direct sum of the k {\displaystyle k} -th exterior powers of V , {\displaystyle V,} and this makes the exterior algebra a graded algebra .
is the linear combination of vectors and such that = +. In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).