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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?

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  • $\begingroup$ Hint: can you form two equations in four unknowns look at the associated kernel? $\endgroup$ – Sean Roberson Aug 10 '17 at 19:39
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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 $\begin{bmatrix}2 \\ 0 \\ 1 \\ 1\end{bmatrix}$.

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