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A2 - Math 235 Assignment 2 Due Wednesday Sept 29th 1 For...

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Unformatted text preview: Math 235 ' Assignment 2 Due: Wednesday, Sept 29th 1. For each of the following linear transformations, determine a geometrically natural basis B and determine the matrix of the transformation with respect to B. a) The projection proj(3,2) : R2 —> R2 onto the line i = t [3] , t E R. ——1 b) The reflection refi(_1,1,1) : R3 —+ R3 over the plane with normal vector 73 : 1 1 2. Let V be a finite dimensional vectors Space and let S, B, and C be bases for V. Suppose that P is the change of basis matrix from S to B, and that Q is the change of basis matrix from S to C. Let L : V —> V be a linear operator. Express the matrix [Llc in terms of [L]B, P, and Q. 3. Let V be an n dimensional vector space with basis B and let S be the vector space of all linear operators L : V —+ V. Define T : S —+ M(n, n) by T(L) = [L]B. Prove that T is a linear mapping. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems, and submit written answers to the questions along with the rest of your assignment. For questions that involve a set of vectors, enter each vector separately, giving it a name (e.g., 51, 52, s3) and then construct any required matrices from these vectors (e.g., A = [51 52 53]). This will make it easier for you to work with the various elements. Matrix Representation of a Linear Mapping Let L : M(3, 2) ——+ P5 be the linear mapping (a1~— 2&2 + 5&3 + 3&4 — 3G5 + a6):c5 + (4a1 —- a4 + 3&5 —~ 3a6)x4 L :1 :2 _ + (~5a2 — 3a3 + 2m; + (15 —~ 2a6)x3 a3 a4 + (50,1 + 5a2 — 4% —~ 5a4 — (15 —~ 2a6)$2 5 6 + (4a1 — 3% + 2a5)$ + (a1 + 2a2 + a3 + 2a4 + a5 —~ 2%) (a) Find A, the matrix representation of the linear mapping L, with respect to the basis, 8 for M(3, 2) and the basis C for P5, where 10 10 —1—1 00 02 12 8: 00,10, 003—1~1,~22,01 01 10 —1—1 00 ~20 20 C: {$5+1,:z:5+x3+:r,—:c5~—$4—x—1,~—$3—x2,2:c4—~2$3+2x2—2:c,:r5+2:c4+:r2+2:r} (b) Use A to find [L(v)]c where [Vlg = (—2,6, —2, —6,4, «2). ...
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