PS12.pdf - Problem Set 12 Orthogonal Projections and...

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Problem Set 12 - Orthogonal Projections and Orthonormal Bases Learning Objectives: You should understand how we use the dot product to define geometric ideas like the length of a vector and the angle between two vectors in R n . You should understand what it means for a collection of vectors to be orthonormal , and you should be able to determine whether given vectors are orthonormal. You should understand how orthogonal projection onto a subspace V of R n is defined. You should be familiar with Theorem 5.1.5 as a way of computing the projection of a vector ~x onto a subspace V of R n , and you should understand why it’s critical here to use an orthonormal basis of V . You should understand the orthogonal complement of a subspace of R n and be able to visualize it in simple cases (in R 2 or R 3 ). The first two problems below are warmup; you need not turn them in. W1. If you’re feeling rusty on this, please read the “Basics of Vectors and Matrices” handout or Appendix A of Bretscher. (a) Find the angle between 1 2 1 and 1 - 1 1 . (b) If ~v = 1 0 1 - 2 , find k ~v k , the length of ~v

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