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).

Let a = \left( {{a_1},{a_2},0} \right) and b = \left( {{b_1},{b_2},0} \right) be tow 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.