Linear map

Definition

Given two vector spaces $(U,F)$ and $(V,F)$ (it is important that they are over the same field) we say that a map,

$$T:(U,F)\rightarrow(V,F)$$ or simply $$T:U\rightarrow V$$ (because mathematicians are lazy), is a linear map if:

$$\forall \lambda,\mu\in F$$ and $$\forall x,y\in U$$ we have $$T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)$$

Notations

Some authors use

$$L$$ for a linear map.

Because linear maps can often (always if $U$ and $V$ are finite dimensional) be represented as a matrix sometimes the notation

$$Tv$$ is used instead of $$T(v)$$

Between a basis

The Change of basis matrix ought to be denoted

$$[Id]_A^B$$ where $A$ is the source basis and $B$ is the target, see this page for a tour of notation and the use of $$[\cdot]_A^B$$

Homomorphism, isomorphism and isometry

A linear map is a vector space homomorphism, if it is a bijection then it is invertible, but the word isomorphism should be used sparingly, to avoid confusion with linear isometries which ought to be called "isometries"

Using the prefix "linear" avoids this, eg:

• Linear homomorphism
• Linear isomorphism
• Linear isometry

Categories

The set of all linear maps from $(U,F)$ to $(V,F)$ is often denoted by

$$\mathcal{L}(U,V)$$ or $$\text{Hom}(U,V)$$

See also

Example comparing bilinear to linear maps