This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 152: Solutions to assignment 7 Problem 1: Let T and S denote the linear transformations of R 2 which reflect vectors in the xaxis, and rotate vectors by 30 degrees clockwise. a) Find matrices representing S and T . Solution : We need to find the image of the coordinate vectors in each case. For the reflection T , the reflection of (1 , 0) is (1 , 0) again, as this point lies on the axis of reflection. The reflection of (0 , 1) is (0 , 1), so finally we get T = 1 1 . As for S , we use the formula for an anticlockwise rotation from the notes (it may be useful to simply memorize the formula). This gives S = cos( ) sin( ) sin( ) cos( ) , with = 30 degrees (since we are rotating clock wise). b) Find matrices representing ST and TS . Solution : This is a direct computation; ST = cos( ) sin( ) sin( ) cos( ) , while TS = cos( ) sin( ) sin( ) cos( ) . Problem 2: Suppose T is a linear transformation from R 2 to R 3 , given by the matrix 1 1 1 2 . If L is a line in R 2 , passing through the point (1 , 1) and in the direction of the vector ~v = (1 , 2 , 1), then find the image of the line L under T . (Note: the vectors in this problem and the problem below have been written as row vectors to save space.) Solution : In general, a line is given by a parametric equation p + sv , where p is the initial point, v is a direction vector, and s is a parameter. The question asks us to describe the image under...
View
Full
Document
 Spring '08
 Caddmen
 Transformations, Vectors, Matrices

Click to edit the document details