Let $${\displaystyle V}$$ and $${\displaystyle W}$$ be vector spaces over the same field $${\displaystyle K}$$. A function See more

• A prototypical example that gives linear maps their name is a function $${\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto cx}$$, of which the graph is a line through the origin.
• More generally, any homothety See more

The composition of linear maps is linear: if $${\displaystyle f:V\to W}$$ and $${\textstyle g:W\to Z}$$ are linear, then so is their See more

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require … See more

Given a linear map which is an endomorphism whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] … See more

If $${\displaystyle V}$$ and $${\displaystyle W}$$ are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from $${\displaystyle V}$$ to $${\displaystyle W}$$ can be represented by a See more

A subtler invariant of a linear transformation $${\textstyle f:V\to W}$$ is the cokernel, which is defined as $${\displaystyle \operatorname {coker} (f):=W/f(V)=W/\operatorname {im} (f).}$$
This is the dual notion to the kernel: just as the kernel is a … See more