Proofs for linear maps – "Math for Non-Geeks"

Fehler: Aktuelle Seite wurde in der Sitemap nicht gefunden. Deswegen kann keine Navigation angezeigt werden 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 $f\colon V\to W$, 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 $f\colon V\to W$ 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: $\forall v,\,w\in V:\quad f(v+w)=f(v)+f(w)$
2. homogeneity: $\forall v\in V\,\forall \lambda \in K:\quad f(\lambda \cdot v)=\lambda \cdot f(v)$

The map to zero

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

$f\colon \mathbb {R} \to \mathbb {R} ^{3},\quad x\mapsto {\begin{pmatrix}0\\0\\0\end{pmatrix}}$

An example in $\mathbb {R} ^{2}$

We consider an example for a linear map of $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$:

$f\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}$ with $f{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}x_{1}+x_{2}\\x_{1}-5x_{2}\end{pmatrix}}$

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.

Math for Non-Geeks: Template:Aufgabe

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

$(f+g)(m)=f(m)+g(m)$

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

$(\lambda \cdot f)(m)=\lambda \cdot f(m)$

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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