Exercise 1.2.17

Question

Let $V=\left\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in F\right\}$, where $F$ is a field. Define addition of elements of $\mathrm{V}$ coordinatewise, and for $c \in F$ and $\left(a_{1}, a_{2}\right) \in \mathrm{V}$, define
$$
c\left(a_{1}, a_{2}\right)=\left(a_{1}, 0\right)
$$
Is $\mathrm{V}$ a vector space over $F$ with these operations? Justify your answer.

Jephian C.-H. Lin · Accepted Answer

No. Since $0(a_1,a_2)=(a_1,0)$ is the zero vector but this will make the zero vector not be unique, it cannot be a vector space.

Rene Girard · Answer

No. (VS8) is not satisfied. Let $c,d \in F$ and $x=(a_1,a_2)$ then $(c+d)x=(a_1,0)$ but $cx +dx = (a_1,0)+(a_1,0) =(2 a_1,0)$ which is NOT equal to $(a_1,0)$.

Exercise 1.2.17

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Exercise 1.2.17

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