The problem is Find an orthonormal{w_1, w_2} for the orthogonal complement of the subspace U given by
2x+y-2z=0
2y-x+2z+w=0
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2026-03-25 15:58:21.1774454301
Got stuck on Finding an orthonormal basis for the orthogonal complement
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To find (x, y, z, w) satisfying 2x+y-2z= 0 and 2y-x+2z+w=0 add the equations and eliminating z: x+ 3y+ w= 0. w= -x- 3y. So (x, y, z, -x- 3y) is a general solution. The set of all solutions is a three-dimensional subspace with basis {(1, 0, 0, -1), (0, 1, 0, -1), (0, 0, 1, 0)}.
The orthogonal complement is a one-dimensional subspace of vectors that have 0 dot product with each of those. That is, if (x, y, z, w) is in the orthogonal subspace, x- w= 0 y- w= 0 z= 0. That is (x, x, 0, x)= x(1, 1, 0, 1). The length of that is $x\sqrt{3}$.
Since that is one dimensional, "orthogonal" is not relevant. in order to be "normalized" x must be $\frac{1}{\sqrt{3}}= \frac{\sqrt{3}}{3}$. The orthogonal subspace is has orthonormal basis $\left(\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3}, 0, \frac{\sqrt{3}}{3}\right)$.