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3 Nov Math 523 Notes

3 Nov Math 523 Notes - M ath 423 Notes 3 November 2010...

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Math 423 Notes 3 November 2010 Section 4.5 Let’s look at the special product: A T A 1. What is the rank (A T A)? rank(A), page 212 2. What is the R(A T A)? R(A T ) ***not doing this proof in class, may be used on a quiz or test*** 3. What is the N(A T A)? 4.4.6 tells us that if the dimensions are the same then, N(A T A) = N(A) Use: N(B) N(AB), basically what we are trying to show is if you picture N(AB), N(B) would be a subset or equal to N(AB). Want to show that x ϵ N(B) x ϵ N(AB) If x ϵ N(B) –what information can I take from the definition of N(B)? .... B x = 0 (definition of nullspace) B x = 0 AB x = A 0 AB x = 0 , therefore, N(B) N(AB) Now we can say: N(A) N(A T A) Looking at pg 198, 4.4.6, we have the assumption required to use 4.4.6. If the dimensions of these two spaces are equal, dim N(A) = dim N(A T A), then we can make the conclusion that the vector spaces are equal, N(A) = N(A T A) Looking at pg 199: dim N(A) = n - r dim N(A) = n – rank (A) dim N(A) = dim N(A T A) pg 199, 4.4.8 and N(A) = N(A T A) Given this as a fact: R(A T A) = R (A T ) Consider: A x = b , may/may not be consistent A T

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