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Exercise 14

Avg difficulty 3.0

Verify that this map h ⁣:R3→Rh\colon\mathbb{R}^{3}\to\mathbb{R}

(xyz)  ↦  (xyz)∙(3−1−1)=3x−y−z\begin{pmatrix}x\\ y\\ z\end{pmatrix}\;\mapsto\;\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mathbin{\smash{\bullet}}\begin{pmatrix}3\\ -1\\ -1\end{pmatrix}=3x-y-z

is linear. Generalize.

Hefferon, Linear Algebra, Maps Between Spaces, Definition, Exercise 14

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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.