Math 250a Problem Set 8 Solutions

Math 250a Problem Set 8 Solutions - Mathematics 250 Fall...

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Unformatted text preview: Mathematics 250 Fall 2006 Selected Solutions Assignment #8 Proof Practice # 8 : Suppose that A is an invertible k × k matrix, and that B is another k × k matrix such that || B- A || < 1 || A- 1 || . Show that B is invertible, and that || B- 1 || ≤ || A- 1 || 1-|| B- A |||| A- 1 || . Response: Following the suggestion in the problem statement, we write B = A ( I- A- 1 ( A- B )). We recall the standard formula ( ST )- 1 = T- 1 S- 1 for the inverse of a product of invertible matrices. This formula shows how to compute the inverse of a product, and in particular, shows that the product of invertible matrices is also invertible. From our formula above for B , we see that this implies that if A and I- A- 1 ( A- B ) = I- Z are invertible, then B will also be invertible. We are given that A is invertible; thus it only remains to establish that I- Z is invertible. From Proof Practice #7, we know that if Z is a matrix of norm less than one, then I- Z is invertible. We can estimate that || Z || = || A- 1 ( A- B ) || ≤ || A- 1 || || A- B || < || A- 1 || 1 || A- 1 || = 1 . The first inequality follows from submultiplicativity, and the second, strict, inequality, follows by the given assumption on || B- A || = || A- B || . Thus, I- Z is indeed invertible, and therefore so is B . Furthermore, Proof Practice #7 also gives us an estimate for || ( I- Z ) || . Applying this and again using submultiplicativity, we can say that || B- 1 || = || ( I- Z )- 1 A- 1 || ≤ || ( I- Z )- 1 || || A- 1 || ≤ 1 1- || Z || || A- 1 || . ( Best ) Since || Z || ≤ || A- 1 | || A- B || < 1, we see that 1- || Z || ≥ 1- || A- 1 || || A- B || = 1- || B- A || || A- 1 || > 0. Taking reciprocals, we see that 1 1-|| Z || ≤ 1 1-|| A- 1 || || A- B || . Using this in the estimate (Best), we finally conclude that || B- 1 || ≤ 1 1- || A- 1 || || A- B || || A- 1 || = || A- 1 || 1- || B- A || || A- 1 || , as desired. AS8.1 : a) List the standard basis for Λ 3 ( R 5 ). b) Compute 1 1 1 ∧ 1- 1 3 2 ∧ 1- 1- 2 . c) Find the Gram matrix of the three vectors of part b), and compute the volume of the parallelopiped spanned by them, and the areas of its faces....
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Math 250a Problem Set 8 Solutions - Mathematics 250 Fall...

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