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1. Suppose v and u 6= 0 are vectors of the same dimension. Verify that v − proju(v) is orthogonal to u. Hint: Use the dot product! Now let B = ( 1 1 −1 1 −1 1 −1 1 1 ) be a set of three vectors in R 3 . We will modify this set of vectors to make it nicer in some respects. 2. Use the idea in 1 to modify the second vector in B to make it orthogonal to the first vector in B. 3. Modify the third vector in B to make it orthogonal to both the first and second vectors in B. 4. Modify the first vector in B and your modified vectors from 2 and 3 to have length one. 5. What might your final collection of modified vectors from 4 be good for ?

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Matrix Algebra Problems
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