136A4S - )-cos(2 ) sin(2 ) cos(2 ) cos(2 ) + sin(2 ) sin(2...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ASSIGNMENT 4 for SECTION 001 Solutions Questions from the textbook Section 3-3: B1 [3 marks] , B3 (c) [1 mark] , B4 [4 marks] , B5 (a) [2 marks] , B6 (a) [3 marks] , D4 [3 marks] and D6 [2 marks] In addition to the questions listed here, do the MATLAB question from Assignment 4 for the other sections; this may be found under Content > Assignments > Assignment 4 [2 marks] Other questions 1. [6 marks] Given an angle θ , let R θ denote the linear transformation reflecting vectors across the line formed by rotating the x -axis counterclockwise by θ . Find all conditions on α and β such that R α and R β commute. R θ reflects vectors across the line x sin θ - y cos θ = 0; that is, across the vector r = ± cos θ sin θ ² . Let e 1 and e 2 denote the columns of the 2 × 2 identity matrix. We have R θ ( e 1 ) = e 1 - 2 ( e 1 - proj r e 1 ) = 2proj r e 1 - e 1 = 2 ³ e 1 · r k r k 2 ´ r - e 1 = ± 2 cos 2 θ - 1 2 cos θ sin θ ² = ± cos(2 θ ) sin(2 θ ) ² . Similarly, R θ ( e 1 ) = ± sin(2 θ ) - cos(2 θ ) ² . Thus R θ has standard matrix ± cos(2 θ ) sin(2 θ ) sin(2 θ ) - cos(2 θ ) ² . Let A and B denote the standard matrices for R α and R β , respectively. Then AB = ± cos(2 α ) cos(2 β ) + sin(2 α ) sin(2 β ) cos(2 α ) sin(2 β ) - sin(2 α ) cos(2 β ) sin(2 α ) cos(2
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: )-cos(2 ) sin(2 ) cos(2 ) cos(2 ) + sin(2 ) sin(2 ) = cos (2( - ))-sin (2( - )) sin (2( - )) cos (2( - )) . Thus R R is a rotation by 2( - ). Similarly, R R is a rotation by 2( - ). These rotations are equal when and are congruent modulo / 2. 2. [4 marks] Let K : R m R n and L : R n R m be linear transformations, and let m < n . Prove that the reduced row-echelon form of the standard matrix of K L is not an identity matrix. Let K and L have standard matrices A and B , respectively. AB , the standard matrix of K L , is m m . Now the reduced row-echelon form of AB is the m m identity matrix if, and only if, Nul( AB ) = { } . Since m < n , B has at least one free variable. Hence there exists a nonzero vector x R m such that B x = , and AB x = . Thus Nul( AB ) 6 = { } ....
View Full Document

This note was uploaded on 03/19/2009 for the course M m136 taught by Professor Fokshuen during the Spring '09 term at Waterloo.

Ask a homework question - tutors are online