I've been given a linear space made by 2 equations $U = \{\ {(x,y,z,w) ∈ R^4: x - y + z = 0 \land y - z + w = 0} \}\ $
and I have to get the orthonormal base of this equation and what I did to to that is I turned it into this matrix $\begin{pmatrix}1&-1&1&0\\ 0&1&-1&1 \\ \end{pmatrix}$ and what I did was the null space of this matrix and with its vectors I divided each vector for its magnitude giving me in the end $\begin{pmatrix}0\\ \frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\\ 0 \\ \end{pmatrix}$ $\begin{pmatrix}\frac{\sqrt{3}}{3}\\ -\frac{\sqrt{3}}{3}\\ 0\\ \frac{\sqrt{3}}{3} \\ \end{pmatrix}$
but the answer is $(-\frac{\sqrt{10}}{5},-\frac{\sqrt{10}}{10},\frac{\sqrt{10}}{10},-\frac{\sqrt{10}}{5})$ so what did I do wrong?