Question

V =((x,y,0)T ∈R3 :x,y∈R).Using the definition of the subspace, show that V is a subspace of the vector space R3.Using the definition of the subspace, show that V is a subspace of the vector space R3.

Answer 1

Let us show that

$V =\{(x,y,0)\in\R^3 :x,y∈\R\}$

is a subspace of $\R^3.$

Let $(x,y,0),(z, t,0)\in V,\ a\in \R.$

Then $(x,y,0)+(z, t,0)=(x+z,y+t,0)\in V,$

$a(x,y,0)=(ax,ay,0)\in V.$

Therefore, $V$ is a subspace of $\R^3.$