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 𝑉,𝑊 ∈𝖵𝖾𝖼𝗍𝕂 a mapping 𝑓 :𝑉 →𝑊 is linear iff for any 𝜆,𝜇 ∈𝕂 and ⃗𝐯,⃗𝐮 ∈𝑉 linalg

𝑓(𝜆⃗𝐯+𝜇⃗𝐮)=𝜆𝑓(⃗𝐯)+𝜇𝑓(⃗𝐮)

It follows that 𝑓(⃗𝟎) =⃗𝟎. A linear map is an example of a Module homomorphism.

Geometric interpretation

If a map 𝑓 :ℝ𝑛 →ℝ𝑚 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 𝑇 ∈𝖵𝖾𝖼𝗍𝕂(𝑈,𝑉) is epic iff it is surjective iff im⁡𝑇 =𝑉
• A linear map 𝑇 ∈𝖵𝖾𝖼𝗍𝕂(𝑈,𝑉) is monic iff it is injective iff ker⁡𝑇 ={⃗𝟎}
• 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. ↩