Serlo: EN: Proofs for linear maps

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}} We will give here a proof structure that shows how to prove linearity of a map.

General procedure

Recap: Definition of a linear map

We recall that a linear map (or homomorphism) is a structure-preserving map of a $K$-vector space $V$ into a $K$-vector space $W$. That is, for the map $f:V\to W$, the following two conditions must hold:

1. $f$ must be additive, i.e., for $v,w\in V$ we have that: $f(v+w)=f(v)+f(w)$
2. $f$ must be homogeneous, i.e., for $v\in V,\lambda \in K$ we have that: $f(\lambda \cdot v)=\lambda \cdot f(v)$.

So for a linear map it doesn't matter if we first do the addition or scalar multiplication in the vector space $V$ and then map the sum into the vector space $W$, or first map the vectors $v,w$ into the vector space $W$ and perform the addition or scalar multiplication there, using the images of the map.

Proving that a map is linear

The proof that a map is linear can be done according to the following structure. First, we assume that a map $f:V\to W$ is given between vector spaces. That is, $V$ and $W$ are $K$-vector spaces and $f$ is well-defined. Then for the linearity of $f$ we have to show:

1. additivity: $\mathrm{\forall }v,w\in V:f(v+w)=f(v)+f(w)$
2. homogeneity: $\mathrm{\forall }v\in V\mathrm{\forall }\lambda \in K:f(\lambda \cdot v)=\lambda \cdot f(v)$

The map to zero

The map to zero is the map which sends every vector to zero. For instance, the map to zero of $ℝ$ to ${ℝ}^3$ looks as follows:

An example in ${ℝ}^2$

<section begin=aufgabe_linearität_R^2 /> We consider an example for a linear map of ${ℝ}^2$ to ${ℝ}^2$:

$f:{ℝ}^2\to {ℝ}^2$ with $f\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}{x}_1+x_2\\ x_1-5x_2\end{array}\right)$

Mathe für Nicht-Freaks: Vorlage:Aufgabe <section end=aufgabe_linearität_R^2 />

A linear map in the vector space of sequences

Next, we consider the space of all sequences of real numbers. This space is infinite-dimensional, because there are not finitely many sequences generating this sequence space. But it is a vector space, as we have shown in the chapter about sequence spaces.

<section begin=folgenraum_abbildung_linear /> Mathe für Nicht-Freaks: Vorlage:Aufgabe <section end=folgenraum_abbildung_linear />

Abstract example

In this chapter, we deal with somewhat more abstract vectors. Let $M,N$ be arbitrary sets; $K$ a field and $V$ a $K$-vector space. We now consider the set of all maps/ functions of the set $M$ into the vector space $V$ and denote this set with $\text{Fun}(M,V)$. Furthermore, we also consider the set of all maps of the set $N$ into the vector space $V$ and denote this set with $\text{Fun}(N,V)$. The addition of two maps is defined for $f,g\in \text{Fun}(M,V)$ by

Die scalar multiplication is defined for $\lambda \in K$ via

Analogously, we define addition scalar multiplication for $\text{Fun}(N,V)$.

We now show that the precomposition with a mapping $t\in \text{Fun}(N,M)$ is a linear map from $\text{Fun}(M,V)$ to $\text{Fun}(N,V)$.

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