lina-30-10-09-p1

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

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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

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Unformatted text preview: R 3 which satisfy u Â· x = 3, where u = 1 1 1 . 4. Show that if V,W âŠ‚ R n are linear subspaces then V âˆ© 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) Â± 2 1 Â² , Â±-14-7 Â² (b) Â± 1-1 Â² , Â±-1 2 Â² (c) 1-1 1 , 1 1 1 , 2 2 7. Show that if the k vectors v 1 , v 2 Â·Â·Â· v k âˆˆ R n are all non-zero and mutually orthogo-nal, i.e., v i Â· v k = 0 if i 6 = k , then they are linearly independent. 2...
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lina-30-10-09-p1 - R 3 which satisfy u Â x = 3 where u = 1...

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