(Worth 10 points total) Let \(V=\mathbb{R}^2\), with the fol…
(Worth 10 points total) Let \(V=\mathbb{R}^2\), with the following addition and scalar multiplication operations on it: \(\begin{align*} \begin{bmatrix} x_1\\x_2 \end{bmatrix} + \begin{bmatrix} y_1\\y_2 \end{bmatrix} &= \begin{bmatrix} x_1+y_1\\x_2+y_2 \end{bmatrix} \text{(that is, the usual addition),} \\\alpha \odot \begin{bmatrix} x_1\\x_2 \end{bmatrix} &= \begin{bmatrix} \max\{\alpha, x_1\}\\\max\{\alpha, x_2\} \end{bmatrix},\end{align*}\) where \(\max\{a,b\}\) means to take the larger of the two numbers \(a\) and \(b\). List TWO vector space axioms that this space \((V, +, \odot)\) violates. For each one, give an example that demonstrates the violation. Write your final answers in the text box below (just name the axioms- your full answers with examples will be submitted to Gradescope as soon as you submit on Canvas).