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

Example:
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 ${V_3}\left( F \right)$.

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