Linear algebra MOC

Linear map

A linear map¹ is a structure-preserving map of vector spaces. That is, given two vector spaces over the same field a mapping is linear iff for any and linalg

It follows that . A linear map is an example of a Module homomorphism.

Geometric interpretation

If a map is interpreted as the warping of space, the above rules are equivalent to the following

• The origin remains in place
• Grid lines remain evenly spaced
• Grid lines remain parallel

Properties

Some of these properties apply for a more general Module homomorphism

• A linear map is epic iff it is surjective iff
• A linear map is monic iff it is injective iff
• A linear map is an isomorphism iff it is bijective iff it is epic and monic
• Rank-nullity theorem

Related

• Linear kernel
• Bounded operator

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Footnotes

1. variously called a linear transformation, linear operator, linear function, linear morphism. ↩