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Exercise 10

Avg difficulty 3.0

Prove that this is not a vector space: the set of two-tall column vectors with real entries subject to these operations.

(x1y1)+(x2y2)=(x1−x2y1−y2)r⋅(xy)=(rxry)\begin{pmatrix}x_{1}\\ y_{1}\end{pmatrix}+\begin{pmatrix}x_{2}\\ y_{2}\end{pmatrix}=\begin{pmatrix}x_{1}-x_{2}\\ y_{1}-y_{2}\end{pmatrix}\qquad r\cdot\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}rx\\ ry\end{pmatrix}

Hefferon, Linear Algebra, Vector Spaces, Definition and Examples, 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.