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ass3 - ASSIGNMENT 3 MATH 270 WINTER 2010 DUE 5PM...

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ASSIGNMENT 3 MATH 270, WINTER 2010. DUE: 5PM, MARCH 11, 2010. Assignments must have your name, student number, course number and TA’s name on the first page. Please submit them by placing them in the box marked “Assignments” outside the Maths Office (Burnside Hall, 10th floor). Show all working and justifications! (1) (a) (i) Is transposition A A T a linear transformation? [ 1 mark ] (ii) Write down the matrix T representing transposition on the vector space of 2 × 2 matri- ces. Use the basis given by the four matrices 1 0 0 0 , 0 1 0 0 , 0 0 1 0 and 0 0 0 1 . [ 3 marks ] (iii) Why must T 2 = I ? [ 1 mark ] (b) (i) Find the matrix R z representing a 90 anticlockwise rotation about the z -axis in R 3 . Use the standard basis. [ 2 marks ] (ii) Repeat for the matrix R x representing a 90 anticlockwise rotation about the x -axis in R 3 . [ 2 marks ] (iii) Show that R x R z = R z R x (rotations in R 3 don’t commute). [ 1 mark ] [ Total: 10 marks ] (2) The set P of all polynomials in x with real coefficients is a real vector space of infinite dimension. (a) Show that the infinite set 1 , x , x 2 , x 3 ,... is a basis of P by demonstrating linear indepen- dence and arguing why it spans P . [ 1 mark ] (b) Construct the (infinite) matrix D representing the linear transformation which takes p ( x ) to its derivative p ( x ) . Use the basis given above. [ 1 mark ] (c) Consider the function on P taking p ( x ) to xp ( x ) . Show that this is a linear transformation, and find the matrix M representing it, again with respect to the above basis. [ 2 marks ] (d) Compute DM - MD

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