Find an orthonormal basis for the subspace Gram-Schmidt

Find an orthonormal basis for the subspace $W = \{[x_1, x_2, x_3, x_4] \ | \ x_1 = x_2 + 2 x_3, \ x_4 = -x_2+x_3\}$

I know how to apply the Gram-Schmidt process but i couldn't form a matrix. What should be my approach?

• Hint: can you form two equations in four unknowns look at the associated kernel? – Sean Roberson Aug 10 '17 at 19:39

1 Answer

Hint: $$\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{bmatrix} = \begin{bmatrix}x_2+2x_3 \\ x_2 \\ x_3 \\ -x_2+x_3\end{bmatrix} = x_2 \begin{bmatrix}1 \\ 1 \\ 0 \\ -1\end{bmatrix} + x_3 \begin{bmatrix}2 \\ 0 \\ 1 \\ 1\end{bmatrix}.$$ All you need is to find two orthonormal vectors to represent $\begin{bmatrix}1 \\ 1 \\ 0 \\ -1\end{bmatrix}$ and $\begin{bmatrix}2 \\ 0 \\ 1 \\ 1\end{bmatrix}$.