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Linear map

Definition

A linear transformation, or linear map, is a function $T:V \to W$ between vector spaces over a field $\mathbb{F}$ that is linear, i.e. $\forall v,w \in V; \alpha \in \mathbb{F}$:

• $V(v+w) = V(v) + V(w)$
• $V(\alpha v) = \alpha V(v)$

In other words, it is a homomorphism between vector spaces.

Criterion

For all $v,w \in V; \alpha \in \mathbb{F}$:

• $V(\alpha v + w) = \alpha V(v) + V(w)$

Properties

By picking bases for $V$ and $W$ we can represent $T$ as a matrix.

The linear maps $T: V \to W$ form a vector space.

Types

If $V=W$ we say $T$ is an endomorphism.

If $T$ is bijective we say $T$ is an isomorphism.

See also

• Ring homomorphism

all these pages adapted with probably insufficient credit from my university's lecture notes