HW3_solutions 167 - Math 167 homework 3 solutions October...

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: Math 167 homework 3 solutions October 20, 2010 2.1.22: For which right-hand sides (find a condition on b 1 , b 2 , b 3 ) are these systems solvable? (a) 1 4 2 2 8 4- 1- 4- 2 x 1 x 2 x 3 = b 1 b 2 b 3 r 2- 2 r 1 and r 3 + r 1 give the conditions b 2- 2 b 1 = 0 and b 3 + b 1 = 0 for solvability. These conditions are equivalent to b 2 = 2 b 1 and b 3 =- b 1 . (b) 1 4 2 9- 1- 4 x 1 x 2 x 3 = b 1 b 2 b 3 r 2- 2 r 1 and r 1 + r 3 = 0 give the condition b 3 + b 1 = 0 for solvability. This condition is equivalent to b 3 =- b 1 . 2.1.26 The columns of AB are linear combinations of the columns of A . This means: The column space of AB is contained in (possibly equal to) the column space of A . Give an example where the column spaces of A and AB are not equal. Let A = 1 0 0 0 1 0 0 0 1 and AB = B = 1 0 0 0 8 0 0 0 0 . 2.1.30 If the 9 by 12 system Ax = b is solvable for every b , then C ( A ) = R 9 . 1 2.2.24 Every column of AB is a combination of the columns of A . Then the dimension of the column space gives rank( AB ) rank( A ). Prove also that rank( AB ) rank( B ). Note that rank( AB ) = n- dim N ( AB ) , where n is the number of columns of AB and dim N ( AB ) is dimension of the nullspace of AB . It is true that dim N ( AB ) dim N ( B ) , since x N ( B ) implies x N ( AB ). Also, note that)....
View Full Document

This note was uploaded on 01/29/2012 for the course MATHEMATIC math ucdav taught by Professor Wiley during the Fall '11 term at UC Merced.

Page1 / 5

HW3_solutions 167 - Math 167 homework 3 solutions October...

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