Exercise 11

Prove or disprove that $\mathbb{R}^{3}$ is a vector space under these operations.

• $\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix}+\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}\quad\text{and}\quad r\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}rx\\ ry\\ rz\end{pmatrix}$

• $\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix}+\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}\quad\text{and}\quad r\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$