Linear Mapping

linear-algebra

Definition

Linear Projection

A mapping $f:V\to W$ between two vector spaces $V$ and $W$ (over the same scalar field $F$) is called linear, if the following two properties apply to $f$ (for $\mathbf{x},\mathbf{y}\in V$ and $\lambda \in F$):

1. $f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y})$
2. $f(\lambda \cdot \mathbf{x})=\lambda \cdot f(\mathbf{x})$

Thus, a linear mapping is a homomorphism from $V$ to $W$.

Examples: Rotations, Mirroring, Projections, …

Linear Combination

Another property of linear mappings is that they can be represented as linear combinations:

$$f({\lambda }_1\cdot {\mathbf{x}}_1+{\lambda }_2\cdot {\mathbf{x}}_2+\dots +{\lambda }_k\cdot {\mathbf{x}}_k)={\lambda }_1\cdot f({\mathbf{x}}_1)+{\lambda }_2\cdot f({\mathbf{x}}_2)+\dots +{\lambda }_k\cdot f({\mathbf{x}}_k)$$

Trivially, this is true since $f$ is a homomorphism.

Composition

Trivially, the composition of two linear mappings is again a linear mapping.

$$f\text{linear}\wedge g\text{linear}⟹(f\circ g)\text{linear}\wedge (g\circ f)\text{linear}$$

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 mapping, 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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