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In this exercise, we investigate the composition of functions AT A mapping R(A) to R(A)
Chapter 3, Problem 25(choose chapter or problem)
In this exercise, we investigate the composition of functions AT A mapping R(A) to R(A), pursuing the discussion on p. 167. a. Suppose A is an m n matrix. Show that ATA = In if and only if the column vectors a1, . . . , an Rm are mutually orthogonal unit vectors. b. Suppose A is anm n matrix of rank 1. Using the notation of Exercise 7, show that ATAx = x for each x R(A) if and only if _u__v_ = 1. Use this fact to show that we can write A = uvT, where _u_ = _v_ = 1. Interpret A geometrically. (See Exercise 4.4.22 for a generalization.)
Questions & Answers
QUESTION:
In this exercise, we investigate the composition of functions AT A mapping R(A) to R(A), pursuing the discussion on p. 167. a. Suppose A is an m n matrix. Show that ATA = In if and only if the column vectors a1, . . . , an Rm are mutually orthogonal unit vectors. b. Suppose A is anm n matrix of rank 1. Using the notation of Exercise 7, show that ATAx = x for each x R(A) if and only if _u__v_ = 1. Use this fact to show that we can write A = uvT, where _u_ = _v_ = 1. Interpret A geometrically. (See Exercise 4.4.22 for a generalization.)
ANSWER:Step 1 of 5
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