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Definition: Linear Map

Let \((F, +, \cdot)\) be a field and let \(V\) and \(W\) be vector spaces over \(F\). A map \(\varphi \colon V\longrightarrow W\) is called a linear map, if the following properties are both fulfilled:

• \(\varphi (u\oplus v)=\varphi (u) \oplus \varphi (v)\) for all \(u,v\in V\),
• \(\varphi (sv)=s\varphi (v)\) for all \(s\in F\) and \(v\in V\).

Table of Contents

Explanations: 1 2

1. Definition: Multilinear Map

Mentioned in:

Definitions: 1 2 3 4 5
Parts: 6
Proofs: 7
Propositions: 8

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück