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If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector x ∈ W; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for v ∈ V.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
Let L(V,W) denote the set of all linear maps from V to W (both of which are vector spaces over F). Then L(V,W) is a subspace of W V since it is closed under addition and scalar multiplication. Note that L(F n,F m) can be identified with the space of matrices F m×n in a natural way. In fact, by choosing appropriate bases for finite-dimensional ...
Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is defined to be the intersection W of all subspaces of V that contain S. It is thus the smallest (for set inclusion) subspace containing W. It is referred to as the subspace spanned by S, or by the vectors in S.
When V = W are the same vector space, a linear map T : V → V is also known as a linear operator on V. A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism. Because an isomorphism preserves linear structure, two isomorphic vector spaces are ...
T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S. T is right-invertible, which is to say there exists a linear map S: W → V such that TS is the identity map on W.
Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W. Given a representation Π : G → GL ( V ) {\displaystyle \Pi :G\rightarrow \operatorname {GL} (V)} , we say that a subspace W of V is an invariant subspace if Π ( g ) w ∈ W ...
In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w. The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V , so bilinear forms may be thought of as elements of ( V ⊗ V ) ∗ which (when V is ...