Answer to Question #309022 in Linear Algebra for stuti

Question

Find a basis and the dimension of the subspace W of ℝ^3 where W = {( x, y, z ) ∈ ℝ^3 ∶ x + y + z = 0 }.

Answer 1

From the equality $x+y+z=0$ we find that $z=-(x+y)$. Thus, the space is 2-dimensional. To find the basis we set: $x_1=1,y_1=0$ and $x_1=0,y_1=1$. We receive vectors: $(1,0,-1)$ and $(0,1,-1)$. This is the basis of $W.$