Serlo: EN: Linear continuation

The principle of linear continuation states that every linear map is exactly determined by the images of the basis vectors. It provides an alternative way to characterize a linear map.

Motivation

So far, we have mostly specified linear maps by saying where each vector of a vector space $V$ is mapped. Those are a lot of vectors, e.g. infinitely many for $V={ℝ}^n$. Is there a way to specify the map with less vectors? Perhaps finitely many ones?

For every vector $v\in V$ of our starting vector space we have to provide the information to which vector of the target vector space it should be mapped. Every such vector can be represented within a basis: If $V$ is a $K$-vector space with basis $\{b_1,\dots ,b_n\}$ and $v\in V$, then there are unique coefficients ${\lambda }_1,\dots ,{\lambda }_n\in K$ such that $v=\sum _{i=1}^n{\lambda }_i{b}_i$ holds.

Now, consider a linear map $f:V\to W$ into another $K$-vector space $W$. The basis vectors of $V$ then have images $f(b_1)=:w_1,\dots ,f(b_n)=:w_n\in W$. Now, an important trick follows: we can use these images $w_1,\dots ,w_n$ as building bricks to construct $f(v)$: by linearity (= additivity + homogeneity) of $f$, we have that: Vorlage:Einrücken

This is amazing: For any $v\in V$, the image $f(v)$ can be reconstructed using $w_1,\dots ,w_n$. Than means the information how the (often infinitely) many $v\in V$ are mapped by $f$ can be condensed in specifying only $n$ vectors! For a linear map $f:{ℝ}^3\to {ℝ}^3$, knowing three vectors $w_1,w_2,w_3$ already suffices to know the image of all infinitely many vectors.

The following theorem assures mathematically that this reconstruction works for any finite dimensional vector space:

Properties of the linear continuation

In the following, $V$ and $W$ are two $K$-vector spaces, $\{b_1,\dots ,b_n\}$ is a basis of $V$ and $w_1,\dots ,w_n\in W$ are vectors in $W$. Let $f:V\to W$ be a linear map with $f(b_i)=w_i$ for all $i\in \{1,\dots ,n\}$. Because of the above theorem such a linear map exists and it is unique.

Exercises

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