Answer to Question #326972 in Linear Algebra for Talifhani

Question

Let x, y and z be three vectors in a vector space V

a. Prove that the span{x,y, z} is a subspace of V.

Answer 1

$u,v\in span\left\{ x,y,z \right\} ,a,b\in K:\\u=\alpha _1x+\beta _1y+\gamma _1z\\v=\alpha _2x+\beta _2y+\gamma _2z\\au+bv=a\left( \alpha _1x+\beta _1y+\gamma _1z \right) +b\left( \alpha _2x+\beta _2y+\gamma _2z \right) =\\=\left( a\alpha _1+b\alpha _2 \right) x+\left( a\beta _1+b\beta _2 \right) y+\left( a\gamma _1+b\gamma _2 \right) z\in span\left\{ x,y,z \right\}$