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Exercise 16

Avg difficulty 3.0

Show that for any scalars a1,1,…,am,na_{1,1},\dots,a_{m,n} this map h ⁣:Rn→Rmh\colon\mathbb{R}^{n}\to\mathbb{R}^{m} is a homomorphism.

(x1⋮xn)↦(a1,1x1+⋯+a1,nxn⋮am,1x1+⋯+am,nxn)\begin{pmatrix}x_{1}\\ \vdots\\ x_{n}\end{pmatrix}\mapsto\begin{pmatrix}a_{1,1}x_{1}+\dots+a_{1,n}x_{n}\\ \vdots\\ a_{m,1}x_{1}+\cdots+a_{m,n}x_{n}\end{pmatrix}

Hefferon, Linear Algebra, Maps Between Spaces, Definition, Exercise 16

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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.