6.1 (9,11,19) 6.2(3,15)

6.1 (9,11,19) 6.2(3,15) - Linear Algebra -115 Solutions to...

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Linear Algebra -115 Solutions to Thirteenth Homework Problem 9 (Section 6 . 1) (a) Assume that < x, z > = 0, for all z β , a basis for a finite dimensional V . Also, assume that β = { v 1 , v 2 , . . . , v n } . Then we can write x = a 1 v 1 + a 2 v 2 + . . . + a n v n . Now, we get < x, x > = < x, a 1 v 1 + a 2 v 2 + . . . + a n v n > = a 1 < x, v 1 > + a 2 < x, v 2 > + . . . + a n < x, v n > = 0 , by our assumption and the fact that v 1 , . . . , v n β . But < x, x > = 0 iff x = 0. (b) Assume that < x, z > = < y, z > , for all z β . Then, < x, z > - < y, z > = 0, or that < x - y, z > = 0, for all z β. Part (a) gives that x - y has to be the zero vector, or that x = y . ± Problem 11 (Section 6 . 1) We make both computations together: k x ± y k 2 = < x ± y, x ± y > = < x, x > + < y, y > ± < y, x > ± < x, y > = k x k 2 + k y k 2 ± < y, x > ± < x, y > Adding these two up, we have that k x + y k 2 + k x - y k 2 = 2 k x k 2 +2 k y k 2 . ± Problem
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This note was uploaded on 01/26/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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6.1 (9,11,19) 6.2(3,15) - Linear Algebra -115 Solutions to...

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