Question

Determine whether W = {(x, y,z) | x + y + z + 1 = 0, x, y,z ∈ R} a subspace of R³ or

not?

Answer 1

Let $x=y=z=0$, then $x+y+z+1=1\neq0$, hence the zero vector $(0,0,0)\in R^3$ is not in $W$. Therefore, $W$ is not a subspace of $R^3$.