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Here $ S=\left\{\begin{Vmatrix}x_1\\ x_2\end{Vmatrix}:x_1,x_2\in\mathbb{R}^n\right\} $ and operations defined by equalities $ \alpha\otimes\begin{Vmatrix}x_1\\ x_2\end{Vmatrix}=\begin{Vmatrix}\alpha x_1\\ \alpha x_2\end{Vmatrix}\qquad $ $ \begin{Vmatrix}x_1\\ x_2\end{Vmatrix}\oplus \begin{Vmatrix}y_1\\ y_2\end{Vmatrix}= \begin{Vmatrix}x_1+y_2\\ 0\end{Vmatrix} $ My question: Is $\langle S, \oplus, \otimes\rangle$ a vector space?

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    Is your addition commutative?2012-07-17

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One axiom vector spaces must follow is the existence of a zero vector. You need some element $\vec{0}$ for which $\vec{v}+\vec{0}=\vec{v}$ for all $\vec{v}$. But if $x_2$ is non-zero, your addition operation makes this impossible. So it's not a vector space.