301_F10_hw4_soln

# 301_F10_hw4_soln - AME 301 Homework#4 Solutions | Fall 2010...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: AME 301 Homework #4 Solutions | Fall 2010 1. The rotation matrix is Q = cos sin sin cos : (a) Using the ‘standard basis’ unit vectors x 1 = e 1 = 1 and x 2 = e 2 = 1 as input vectors, the corresponding output vectors are y 1 = Qx 1 = cos sin and x 2 = Qx 2 = sin cos : (b) INSERT SKETCH OF THE 4 VECTORS FOR THETA = 30 DEGREES. (c) By inspection (or by row reduction of A or A T ), the row vectors (and column vectors) are linearly independent. Hence, dim(column space) = 2. A basis for the column space is any two linearly independent vectors in R 2 , say the two columns of A ( a 1 , a 2 ) or the standard basis ( e 1 , e 2 ). (d) We know that dim(null space) + dim(row space) = 2. Since dim(row space = 2), we must have dim(null space) = 0. Hence, the null space contains only the zero vector . (e) Calculate Q 1 using Gauss-Jordan Q I = cos sin 1 0 sin cos 0 1 Multiply row 1 by sin = cos and add to row 2. Noting that cos +sin 2 = cos = 1 = cos , we obtain Q I = cos sin 1 1 = cos sin = cos 1 Next, eliminate q 12 by multiplying row 2 by cos sin and adding to row 1 . Simplifying the result using trigonometric relations, we obtain Q 00 I 00 = cos cos 2 sin cos 1 = cos sin = cos 1 The nal step is to multiply row 1 00 by 1 = cos and row 2 00 by cos . We obtain Q 000 I 000 = 1 0 cos sin 0 1 sin cos 1 Since Q 000 = I , we have I 000 = Q 1 where Q...
View Full Document

## This note was uploaded on 08/25/2011 for the course AME 301 taught by Professor Wu during the Fall '08 term at Arizona.

### Page1 / 4

301_F10_hw4_soln - AME 301 Homework#4 Solutions | Fall 2010...

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

View Full Document
Ask a homework question - tutors are online