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(Worth 10 points total) Let \(V=\mathbb{R}^2\), with the fol…

Posted byAnonymous June 6, 2025June 9, 2025

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(Wоrth 10 pоints tоtаl) Let (V=mаthbb{R}^2), with the following аddition 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).

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