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solutions4bd - Math 33a/2 Quiz 4bd November 8 2007 Name...

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Math 33a/2, Quiz 4bd, November 8, 2007 Name: UCLA ID: 1. Find an orthonormal basis for the subspace V of R 4 , where V = ker 1 0 4 - 3 0 1 2 - 1 . Solution. The matrix 1 0 4 - 3 0 1 2 - 1 is already in rref, so we can easily write down a basis for its kernel: v 1 = - 4 - 2 1 0 , v 2 = 3 1 0 1 . To find an orthonormal basis { u 1 , u 2 } for this space, we apply the Gram-Schmidt process. u 1 = v 1 || v 1 || = 1 21 - 4 - 2 1 0 . To obtain u 2 , we take v 2 - ( v 2 · u 1 ) u 1 and divide by its length. v 2 - ( v 2 · u 1 ) = 3 1 0 1 - ( 3 1 0 1 · 1 21 - 4 - 2 1 0 ) 1
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