Question

Let W={(x,y,z)|x+y+z=0}

a, give a basis for W

b, what is the dimension of W

Answer 1

Let $W=\{(x,y,z)|x+y+z=0\}.$

a) Taking into account that

$W=\{(x,y,z)|x+y+z=0\}=\{(x,y,-x-y)|x,y\in\R\}\\=\{x(1,0,-1)+y(0,1-1)|x,y\in\R\},$

we conclude that the vectors $(1,0,-1)$ and $(0,1,-1)$ form a basis for $W.$

b) Since two vectors form a basis for $W,$ the dimension of $W$ is equal to 2.