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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 .
In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from =), that is, [1] [2]
In particular, a nonzero invariant vector (i.e. a fixed point of T) spans an invariant subspace of dimension 1. As a consequence of the fundamental theorem of algebra, every linear operator on a nonzero finite-dimensional complex vector space has an eigenvector. Therefore, every such linear operator in at least two dimensions has a proper non ...
The row space of this matrix is the vector space spanned by the row vectors. The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors. 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 ...
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 .
Linear subspace A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W. [10] This implies that every linear combination of elements of W belongs to W. A ...
In linear algebra, the quotient of a vector space by a subspace is a vector space obtained by "collapsing" to zero. The space obtained is called a quotient space and is denoted V / N {\displaystyle V/N} (read " V {\displaystyle V} mod N {\displaystyle N} " or " V {\displaystyle V} by N {\displaystyle N} ").
Kernel and image of a linear map L from V to W. The kernel of L is a linear subspace of the domain V. [3] [2] In the linear map :, two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is, = () =.