Answer to Question #277416 in Algebra for Dani

Question

Let V be a vector space over a filed F and x,y,z is an element of V then show that the set of all liner combinations of x,y and z,W=(ax+by+cz:a,b,c are an element of F) is a subspace of V over F. This subspace is called the span of (x,y,z)

Answer 1

Let W={ax+by+cz:a,b,c $\in$ $\mathbb{F}$ }

We want to show that W is a subspace for x,y,z$\in$ V(where V is a vector space over $\mathbb{F}$ ) i.e $\alpha$u+$\beta$v$\in$ W, u,v$\in$ W and $\alpha$ ,$\beta$ $\in$ $\mathbb{F}$ .

Since u$\in$ W => u=$a_1$x + $b_1$y + $c_1$z and v$\in$ W =>$a_2$x + $b_2$y + $c_2$z for$a_1,b_1,c_1,a_2,b_2,c_2 \in \mathbb{F}$

$\alpha u +$ $\beta v$ =$\alpha (a_1x + b_1y + c_1z)$ + $\beta(a_2x +b_2y +c_2z)$

=$\alpha a_1x +\alpha b_1y + \alpha c_1z +\beta a_2x +\beta b_2y+ \beta c_2z$

=$(\alpha a_1 +\beta a_2)x +(\alpha b_1 +\beta b_2)y +(\alpha c_1 + \beta c_2)z$

Since ($\alpha a_1+\beta a_2),(\alpha b_1+\beta b_2),(\alpha c_1+\beta c_2) \in \mathbb{F}$

=>$px+qu+rz \in W ,p,q,r \in \mathbb{F}$

Which implies that $\alpha u + \beta v =px+qu+rz \in W ,p,q,r \in \mathbb{F}$

Hence, W is a subspace of V.