Exercise 1.2.13

Question

Let $\mathrm{V}$ denote the set of ordered pairs of real numbers. If $\left(a_{1}, a_{2}ight)$ and $\left(b_{1}, b_{2}ight)$ are elements of $\mathrm{V}$ and $c \in R$, define
$$
\left(a_{1}, a_{2}ight)+\left(b_{1}, b_{2}ight)=\left(a_{1}+b_{1}, a_{2} b_{2}ight) \quad 	ext { and } \quad c\left(a_{1}, a_{2}ight)=\left(c a_{1}, a_{2}ight) .
$$
Is $\mathrm{V}$ a vector space over $R$ with these operations? Justify your answer.

Jephian C.-H. Lin · Accepted Answer

No. If it's a vector space, we have $0(a_1,a_2)=(0,a_2)$ be the zero vector. But since $a_2$ is arbitrary, this is a contradiction to the uniqueness of zero vector.

Exercise 1.2.13

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Exercise 1.2.13

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