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1. - 1.71. Our proof of the Cauchy-Schwarz inequality, Theorem 1.13, used that when $U$ is a unit vector,  $0 \leq ||V−(U·V)U||^2 = ||V||2 −(U·V)^2$.  Therefore if $U$ is a unit vector and equality holds, then $V = (U · V)U$. Show that equality occurs in the Cauchy Schwarz inequality for two arbitrary vectors $V$ and $W$ only if one of the vectors is a multiple (perhaps zero) of the other vector. 
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$$V = (U · V)U = (\frac{W}{||W||}· V)·\frac{W}{||W||}= (\frac{W·V}{||W||^2})·W$$  As $(\frac{W·V}{||W||^2})$ is a constant, so $V$ is a multiple of $W$.  2. 2.19. -2.19.  Suppose C is an n by n matrix with orthonormal columns. Use Theorem 2.2 to show that  $$||CX|| \leq \sqrt{n} ||X||$$  Use the Pythagorean theorem and the result of Problem 2.17 to show that in fact