Vector Subspace

Let $$V$$ be a vector space over the field $$F$$. Then a non-empty subset $$W$$ of $$V$$ is called a vector space of $$V$$ if under the operations of $$V$$, $$W$$ itself is a vector space over $$F$$. In other words, $$W$$ is a subspace of $$V$$ whenever

$${w_1},{w_2} \in W$$ and $$\alpha ,\beta \in F \Rightarrow \alpha {w_1} + \beta {w_2} \in W$$

Prove that the set $$W$$ of ordered tried $$\left( {{a_1},{a_2},0} \right)$$ where $${a_1},{a_2} \in F$$ is a subspace of $${V_3}\left( F \right)$$.

Let $$a = \left( {{a_1},{a_2},0} \right)$$ and $$b = \left( {{b_1},{b_2},0} \right)$$ be two elements of $$W$$.

Therefore $${a_1},{a_2},{b_1},{b_2} \in F$$ let $$a,b \in F$$ then
\[ \begin{gathered} a\alpha + b\beta = a\left( {{a_1},{a_2},0} \right) + b\left( {{b_1},{b_2},0} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {a{a_1},a{a_2},0} \right) + \left( {b{b_1},b{b_2},0} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {a{a_1} + b{b_1},a{a_2} + b{b_2},0} \right) \in W \\ \end{gathered} \]

Because $$a{a_1} + b{b_1},a{a_2} + b{b_2} \in F$$.