Exercise 1.2.21

Question

Let $\mathrm{V}$ and $\mathrm{W}$ be vector spaces over a field $F$. Let
$$
\mathrm{Z}=\{(v, w): v \in \mathrm{V} 	ext { and } w \in \mathrm{W}\}
$$
Prove that $Z$ is a vector space over $F$ with the operations
$$
\left(v_{1}, w_{1}ight)+\left(v_{2}, w_{2}ight)=\left(v_{1}+v_{2}, w_{1}+w_{2}ight) \quad 	ext { and } \quad c\left(v_{1}, w_{1}ight)=\left(c v_{1}, c w_{1}ight) .
$$

Jephian C.-H. Lin · Accepted Answer

Let $0_V$ and $0_W$ be the zero vector in $V$ and $W$ respectively. Then we have $(0_V,0_W)$ will be the zero vector in $Z$. The other condition could also be checked carefully. This space is called the direct product of $V$ and $W$.

Exercise 1.2.21

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Exercise 1.2.21

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