MATH 211 SPRING 2017, Assignment 3 (due Monday, Feb. 27 (A02) and Tuesday, Feb. 28 (A01) in class) Show your work clearly. Illegible or disorganized solutions may receive no credit. Note: A vector is denoted by boldface (e.g. u) rather than by a hat (e.g. ~u) because the former is nicer to read/write in LATEX. As such, 0 denotes the zero vector. 1. [15: 3 per part] Each of the following asks you to find examples of matrices that satisfy the given conditions. Provide your findings and demonstrate that they satisfy the required conditions. a) Find a pair of matrices A, B such that AB and BA are both defined and AB 6= BA. b) Find a pair of different matrices A and B, neither of which is identity, such that AB and BA are both defined and AB = BA. c) Find a pair of matrices A, B, neither of which is identity, such that A2 = B 3 . d) Find a matrix A, not a diagonal matrix, such that AT = A1 . e) Let O3 be the 3 3 zero matrix. Find a matrix A such that A 6= O3 and A2 6= O3 , but A3 = O3 . [ Note: your matrices must be of size at least 2 2 ] 2. [20] The following concern linear operators on R2 . a) b) c) d) e) f) g) h) i) 3. [1] Find the matrix representations for Ry , reflection across the y-axis, and for Rx , reflection across the x-axis. [1] Find the matrix representation for R/3 , (counterclockwise) rotation about the origin through an angle of /3. [3] The composite mapping Rx R/3 Ry is equivalent to a single rotation, R . Find . [3] Sketch a graph (in the xy-plane) of the unit square with vertices (corners) (0, 0), (1, 0), (1, 1), and (0, 1), and then sketch a separate graph for the image of this square after each of the three steps of the composite mapping in part (c). [1] Find the matrix representation for S2 , a shear by amount 2 in the x direction. [1] Find the matrix representations for T2 , a stretch by a factor 2 in the x direction, and T1/2 , a stretch by a factor 1/2 in the x direction [3] The composite mapping T1/2 S2 T2 is equivalent to a single shear, Ss in the x direction. Find s. [3] Sketch a graph (in the xy-plane) of the unit square with vertices (corners) (0, 0), (1, 0), (1, 1), and (0, 1), and then sketch a separate graph for the image of this square after each of the three steps of the composite mapping in part (g). [4] (Hard) Consider the line through the origin at angle with the x-axis. So, for example, the line y = x is at angle /4. Show that reflection across this line (for any ) is equivalent to first reflecting across the x-axis and then rotating through an angle of 2. [12: 3 per part] For each of the following matrices, find a basis for each of Row(A), Col(A) and Null(A). 1 2 a) A = 2 4 3 6 \u0014 \u0015 1 2 3 4 b) A = 1 2 4 3 c) Your choice of A from 1d). d) Your choice of A from 1e). 1