lina-30-10-09-p1 - R 3 which satisfy u x = 3, where u = 1 1...

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

View Full Document Right Arrow Icon
Linear Algebra Geometry: Sheet 4 Set on Friday, October 30: Questions 2, 3, 4, and 6 1. Use the Cauchy Schwarz inequality | v · w | ≤ k v kk w k to derive the triangle inequality k v + w k ≤ k v k + k w k Hint: express k v + w k 2 in terms of the dot product. 2. Use the Cauchy Schwarz inequality to derive the following relation: For any collection of N real numbers a 1 ,a 2 ,...,a N we have ± a 1 + a 2 + ··· + a N N ² 2 a 2 1 + a 2 2 + ··· + a 2 N N . Hint: Consider v R N with components given by the numbers a 1 ,a 2 , ··· ,a N and find a suitable w R N such that v · w = ( a 1 + a 2 + ··· + a N ) /N . 3. Determine which of the following subsets of R 3 are linear subspaces. (a) The set of all vectors of the form x 0 0 , with x R . (b) The set of all vectors of the form x 1 1 , with x R . (c) The set of all vectors of the form x 1 x 2 x 3 , with x 1 ,x 2 ,x 3 R and x 1 - x 2 = 0. (d) The set of all vectors of the form x 1 x 2 x 3 , with x 1 ,x 2 ,x 3 R and x 1 - x 2 = 3. (e) The set of all vectors x R 3 which satisfy u · x = 0, where u = 1 1 1 . (f) The set of all vectors x
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: R 3 which satisfy u x = 3, where u = 1 1 1 . 4. Show that if V,W R n are linear subspaces then V W is as well a linear subspace of R n . 1 5. Let v 1 = 1 1-1 and v 2 = 1-1 2 and V = span { v 1 , v 2 } , which of the following vectors are in V ? 1 1 , 2 5 , 2-3 9 31-44 6. Are the following sets of vectors linearly dependent or independent? (a) 2 1 , -14-7 (b) 1-1 , -1 2 (c) 1-1 1 , 1 1 1 , 2 2 7. Show that if the k vectors v 1 , v 2 v k R n are all non-zero and mutually orthogo-nal, i.e., v i v k = 0 if i 6 = k , then they are linearly independent. 2...
View Full Document

Page1 / 2

lina-30-10-09-p1 - R 3 which satisfy u x = 3, where u = 1 1...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online