Linear Map

linear-algebra

Definition

Linear Map

A linear map is a function $T:V\to W$ between vector spaces over the same field $F$ that is linear:

$$T(x+y)=T(x)+T(y),T(\alpha x)=\alpha T(x)$$

for all $x,y\in V$, $\alpha \in F$.

If $V=W$, $T$ is a linear operator.

Kernel

Definition

Kernel (Algebra)

The kernel of a group morphism $\phi :G\to H$ is the set of elements $g\in G$ such that $\phi (g)$ maps to the neutral element $e_H$ of $H$:

$$\mathrm{kern}\phi =\{g\in G\mid \phi (g)=e_H\}$$

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The neutral element is $e_H=0$, where $0$ is the zero vector.

Rank

Let $V,W$ be vector spaces and $f:V\to W$ be a linear map, then:

$$\mathrm{rank}(f)=\dim (f(V))$$

Corank

Definition

Corank

The corank of relation $f:V\to W$ is the dimension of the kernel:

$$\mathrm{corank}(f)=\mathrm{def}(f)=\dim (\ker (f))$$

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