This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: ~x ~ y = k ~x + ~ y k 2-k ~x-~ y k 2 2 . Use this fact to prove that a linear transformation that preserves length, must also preserve angles. 9. If A R n n is a matrix, then show that A + A T is symmetric. A matrix H is called skew-symmetric if H T =-H . Show that A-A T is skew-symmetic. Use this to prove that every matrix A can be written in the form A = B + C where B is symmetric and C is skew-symmetric. 10. (hard) The simplex spanned by ~v 1 ,...,~v n R n is the set of vectors of the form t 1 ~v 1 + ...t n ~v n , where t 1 ,...,t n [0 , 1]. The convex hull is the set of points of the form t 1 ~v 1 + + t n ~v n where t 1 + + t n = 1. 1 (a) In two dimensions what is the shape of the simplex and the convex hull of two linearly independent vectors. What is the relationship between their areas. (b) Generalize your answer to n dimensions. 2...
View Full Document