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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 x-axis, 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...
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