Question

Show that the set W = {(a, b, 0) : a, b ∈ F} is a subspace of V3(F).

Answer 1

Let us show that the set $W = \{(a, b, 0) : a, b ∈ F\}$ is a subspace of $V_3(F).$

Let $(a, b, 0),(c, d, 0)\in W.$

Then $(a, b, 0)+(c, d, 0)=(a+c,b+d,0)\in W.$

If $f\in F,$ then $f\cdot(a,b,0)=(fa,fb,0)\in W.$

We conclude that $W = \{(a, b, 0) : a, b ∈ F\}$ is a subspace of $V_3(F).$