Linear map

A linear map¹ is a structure-preserving map of vector spaces. That is, given two vector spaces over the same field $𝑉, 𝑊 \in 𝖵 𝖾 𝖼 𝗍 𝕂$ a mapping $𝑓 : 𝑉 \to 𝑊$ is linear iff for any $𝜆, 𝜇 \in 𝕂$ and $⃗ 𝐯, ⃗ 𝐮 \in 𝑉$ #m/def/linalg

𝑓 (𝜆 ⃗ 𝐯 + 𝜇 ⃗ 𝐮) = 𝜆 𝑓 (⃗ 𝐯) + 𝜇 𝑓 (⃗ 𝐮)

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

Geometric interpretation

If a map $𝑓 : ℝ 𝑛 \to ℝ 𝑚$ 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 $𝑇 \in 𝖵 𝖾 𝖼 𝗍 𝕂 (𝑈, 𝑉)$ is epic iff it is surjective iff $i m 𝑇 = 𝑉$
A linear map $𝑇 \in 𝖵 𝖾 𝖼 𝗍 𝕂 (𝑈, 𝑉)$ is monic iff it is injective iff $k e r 𝑇 = {⃗ 𝟎}$
A linear map is an isomorphism iff it is bijective iff it is epic and monic
Rank-nullity theorem

#state/tidy | #SemBr | #lang/en

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

Linear map

Properties

Related

Footnotes