nsol3 - Review Problems for Exam Three THEORY 1. Use the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Review Problems for Exam Three THEORY 1. Use the Law of Cosines to prove that x T y = || x |||| y || cos ( ) for x,y R 3 . By the Law of Cosines, k x- y k 2 = k x k 2 + k y k 2- 2 k x kk y k cos ( ), where is the angle between x and y . By definition, k x- y k 2 = h x- y,x- y i , which equals h x,x i + h y,y i - 2 h x,y i by properties of inner products. Thus- 2 h x,y i =- 2 k x kk y k cos ( ), so that x T y = h x,y i = k x kk y k cos ( ). 2. Let A be an m n matrix. Identify Rowspace ( A ) , Colspace ( A ) and N ( A ) in terms of the row, column and null spaces of A and/or A t . rs ( A ) = ns ( A ), cs ( A ) = ns ( A T ) and ns ( A ) = rs ( A ) 3. Prove that Rowspace ( A ) = Nullspace ( A ). Let A have rows A T 1 ,...,A T m . Suppose first that x rs ( A ) , so that y T x = 0 for all y rs ( A ). Then in particular A T i x = 0 for each i . It follows that Ax = A T 1 . . . A T m x =- 0 , so that x ns ( A ). Suppose next that x ns ( A ), so that Ax = 0, so that A i x = 0 for each i . Then for any y = c 1 A 1 + + c m A m rs ( A ), y T x = ( c 1 A 1 + ... + c m A m ) T x = c 1 A T 1 x + + c m A T m x = c 1 0 + + c m 0 = 0, so that x rs ( A ) . 4. (a) Prove that N ( A ) = N ( A T A ) for any m n matrix A . (b) Show that rk ( A T A ) = rk ( A T ). (c) Show that A T A- x = A T- b has a solution for any- b R m . (a) Suppose first that x ns ( A ), so that Ax = 0. Then ( A T A ) x = A T ( Ax ) = A T 0 = 0, so that x ns ( A T A ). Suppose next that x ns ( A T A ), so that A T Ax = 0. Then Ax ns ( A T ). Also, Ax cs ( A ). But ns ( A T ) = rs ( A T ) = cs ( A ), so that Ax ns ( A T ) . Thus k Ax k = ( Ax ) T Ax = 0, so that Ax = 0. This means that x ns ( A ). (b) It follows from (a) that A and A T A have the same nullity. A T A is an n n matrix, so that, by the rank plus nullity theorem, rk ( A T A ) = n- null ( A T A ) = n- null ( A ) = rk ( A ). (c) It follows from (a) that rs ( A T A ) = ns ( A T A ) = ns ( A ) = rs ( A ), so that cs ( A T A ) = rs ( A T A ) = rs ( A ) = cs ( A T ). Now A T b cs ( A T ), so that A T b cs ( A T A ), which means that A T b = A T Ax for some x . 5. Show that for any matrix A , the transformation L A is an isomorphism from the Row Space of A onto the Column Space of A . Let A be m n , so that L A maps R n into R m . L A ( v ) = Av is in colspace(A) for any v , so L A maps rowspace(A) into colspace(A). To show that this map is onto, take any y R ( A ) and let y = Ax for some x R m . Now let x = u + v where u rowspace ( A ) and v N ( A ), by Theorem 5.2.3. Then Ax = Au + Av = Au , since Av = 0. Therefore y = Au with u in rowspace(A). To show that this map is one-to-one, suppose that Ax = Au for some x and u in rowspace(A). Then A ( x- u ) = 0, so that x- u N ( A ). But x- u rowspace ( A ), which is N ( A )...
View Full Document

This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

Page1 / 7

nsol3 - Review Problems for Exam Three THEORY 1. Use the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online