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

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online