Graded vector space

Homomorphism of graded vector spaces

Let $𝑓 : 𝑉 \to 𝑊$ be a linear map of $𝑆$ -graded vector spaces over $𝕂$ .

$𝑓$ is graded or grade-preserving iff $𝑓 (𝑉 𝛼) \leq 𝑓 (𝑊 𝛼)$ for all $𝛼 \in 𝑆$ . #m/def/linalg
See $𝖦 𝗋 𝑆 𝖵 𝖾 𝖼 𝗍 𝕂$ .
If $(𝑆, +)$ is a monoid, then $𝑓$ is homogenous of degree $𝛽 \in 𝑆$ iff $𝑓 (𝑉 𝛼) \leq 𝑓 (𝑊 𝛼 + 𝛽)$ and we write $d e g 𝑓 = 𝛽$ . #m/def/linalg
See $𝗀 𝗋 𝔄 𝖵 𝖾 𝖼 𝗍 𝕂$

See Degree operator properties.

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