Serlo: EN: Properties of linear maps

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We consider some properties of linear maps between vector spaces.

Overview

<section begin="Overview"/>A linear map, also called vector space homomorphism, preserves the structure of the vector space. This is shown in the following properties of a linear mapping $f:V\to W$:

• The zero vector is mapped to the zero vector: $f(0)=0$.
• Inverses are mapped to inverses: $f(-v)=-f(v)$.
• Linear combinations are mapped to linear combinations.
• Compositions of linear maps are again linear
• Images of subspaces are subspaces
• The image of a span is the span of the individual image vectors: $f(\mathrm{span}(M))=\mathrm{span}(f(M))$ ($M\subseteq V$ is supposed to be an arbitrary set)<section end="Overview"/>

Zero vector is mapped to the zero vector

The zero vector / origin has a central meaning in our view of vector spaces. And indeed, the origin is sent to the origin by any linear map. Mathematically, "origin" means the neutral element $0$ of addition.

And the $0\to 0$-property can be shown as a mathematical theorem:

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Inverses are mapped to inverses

Another important structure of vector space is that there is an additive inverse $-v$ to every element $v$. We now want to show that inverses are preserved by linear maps.

The statement of this theorem also holds in any abelian group. However, scalar multiplication does not exist there. Hence, the alternative (2nd) version of the proof must be used in this case.

Linear combinations are mapped to linear combinations

Linear mappings preserve the structure of a linear combination and thus map linear combinations in the domain of definition to their corresponding linear combinations in the range of values: <section begin=linear combinations werden auf linear combinations abgebildet /> Mathe für Nicht-Freaks: Vorlage:Satz <section end=linear combinations werden auf linear combinations abgebildet />

Compositions of linear maps are again linear

Let us take two linear maps $f:V_1\to V_2$ and $g:V_2\to V_3$. Both are compatible with the vector space structure and preserve linear combinations. This preservation should also hold for the consecutive execution of both maps $g\circ f:V_1\to V_3$ with $(g\circ f)(v)=g(f(v))$. This is mathematically established by the following theorem:

Subspaces are mapped to subspaces

That linear maps preserve the vector space structure can also be seen in the following property: The images of subspaces of a linear map are again subspaces.

Spans are mapped to spans

Now suppose we have a subset $M\subseteq V$. For this subset, it does not matter whether we first calculate the span and then apply the map or vice versa. This is content of the following theorem:

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