Exercise 1.2.19

Question

Let $\mathrm{V}=\left\{\left(a_{1}, a_{2}ight): a_{1}, a_{2} \in Right\}$. Define addition of elements of $\mathrm{V}$ coordinatewise, and for $\left(a_{1}, a_{2}ight)$ in $\mathrm{V}$ and $c \in R$, define
$$
c\left(a_{1}, a_{2}ight)= \begin{cases}(0,0) & 	ext { if } c=0 \ \left(c a_{1}, \frac{a_{2}}{c}ight) & 	ext { if } c 
eq 0 .\end{cases}
$$
Is $\mathrm{V}$ a vector space over $R$ with these operations? Justify your answer.

Jephian C.-H. Lin · Accepted Answer

No. Because $$(c+d)(a_1,a_2)=\left((c+d)a_1,\frac{a_2}{c+d}\right)$$ may not equal to $$c(a_1,a_2)+d(a_1,a_2)=\left(ca_1+dc_1,\frac{a_2}{c}+\frac{a_2}{d}\right).$$

Rene Girard · Answer

Typo: RHS of second equation in the solution: $c a_1 + d c_1$ should be $c a_1 + d a_1$.

Exercise 1.2.19

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Exercise 1.2.19

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