Exercise 1.2.18

Question

Let $\mathrm{V}=\left\{\left(a_{1}, a_{2}ight): a_{1}, a_{2} \in Right\}$. For $\left(a_{1}, a_{2}ight),\left(b_{1}, b_{2}ight) \in \mathrm{V}$ and $c \in R$, define
$$
\left(a_{1}, a_{2}ight)+\left(b_{1}, b_{2}ight)=\left(a_{1}+2 b_{1}, a_{2}+3 b_{2}ight) \quad 	ext { and } \quad c\left(a_{1}, a_{2}ight)=\left(c a_{1}, c a_{2}ight)
$$
Is $\mathrm{V}$ a vector space over $R$ with these operations? Justify your answer.

Jephian C.-H. Lin · Accepted Answer

No. We have
$$((a_1,a_2)+(b_1,b_2))+(c_1,c_2)=(a_1+2b_1+2c_1,a_2+3b_2+3c_2)$$
but
$$(a_1,a_2)+((b_1,b_2)+(c_1,c_2))=(a_1+2b_1+4c_1,a_2+3b_2+9c_2).$$

Rene Girard · Answer

No. We have

$(a_1,a_2) + (b_1,b_2) = (a_1 + 2 b_1, a_2 + 3 b_2)$

but  $(b_1,b_2) + (a_1,a_2) = (b_1 + 2 a_1, b_2 + 3 a_2)$

so (VS1) (commutativity of addition) is not satisfied.

Exercise 1.2.18

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Exercise 1.2.18

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