# Is $\langle S, \oplus, \otimes\rangle$ a vector space?

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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Please put the entire question in the body of the post. Try and tell us what you tried or didn't try to do. – Asaf Karagila Jul 17 '12 at 16:27
Also, please don't put things like "Prove your answer" in the title. The title is meant to show at a glance what the question is about. When you choose a title, imagine how it will look in the question list on the main page. – joriki Jul 17 '12 at 16:30
Is your addition commutative? – David Mitra Jul 17 '12 at 16:37

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.

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