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Math 369 HW #7 Due at the beginning of class on Friday, October 24 Reading: Sections EE, PEE. Problems: We say that a square matrix M is similar to the square matrix N if there is an invertible matrix A so that M = AN A−1 . Other terms for this: we say that N is a “change of basis” of M or that N is “conjugate” to M . a 0 1. Suppose M = (such a matrix is sometimes called a “scalar matrix”). Are there any 0 a other matrices (i.e., not equal to M ), which are similar to M ? Find one if “yes”, and prove that there can’t be any if “no”. a1 0 . Are there any other 2. Suppose M is a 2 × 2 diagonal matrix, meaning that M = 0 a2 matrices which are similar to M ? Find one if “yes”, and prove that there are none if “no”. 3. Suppose M is similar to N . Is N similar to M ? Prove or give a counterexample. 4. Let f : R2 → R2 be the linear transformation which rotates objects in the plane around the origin by 30 degrees counterclockwise. 0 1 . , (a) Give a matrix F for f with respect to the standard basis 1 0 ( √ ! ) 3 0 2 (b) Give a matrix G for f with respect to the basis . , 1 1 2 (c) Since the matrices in (a) and (b) are two different matrices for the same linear transformation, they must be similar. So find an invertible matrix A so that G = AF A−1 . 5. For each of the following, find the requested matrix and draw a picture showing what the transformation does. * * * * (a) Find a 2 × 2 matrix A for a linear transformation R having the property that R( u) = λ u * * for some constant λ only if u = 0 . (Think of rotations) (b) Find a 2 × 2 matrix B for a linear transformation S having the property that S( u) = λ u a * only when u = for some a ∈ R. (Think of sheering maps) 0 1