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Back to The Permutation Expansion

Exercise 10

Avg difficulty 3.0

Prove that ff is multilinear if and only if for all v⃗,w⃗∈V\vec{v},\vec{w}\in V and k1,k2∈Rk_{1},k_{2}\in\mathbb{R}, this holds.

f(ρ⃗1,…,k1v⃗1+k2v⃗2,…,ρ⃗n)=k1f(ρ⃗1,…,v⃗1,…,ρ⃗n)+k2f(ρ⃗1,…,v⃗2,…,ρ⃗n)f(\vec{\rho}_{1},\dots,k_{1}\vec{v}_{1}+k_{2}\vec{v}_{2},\dots,\vec{\rho}_{n})=k_{1}f(\vec{\rho}_{1},\dots,\vec{v}_{1},\dots,\vec{\rho}_{n})+k_{2}f(\vec{\rho}_{1},\dots,\vec{v}_{2},\dots,\vec{\rho}_{n})

Hefferon, Linear Algebra, Determinants, The Permutation Expansion, Exercise 10

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.