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Math_235-Assig_2-F09

# Math_235-Assig_2-F09 - Math 235 Assignment 2 Due Wednesday...

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Unformatted text preview: Math 235 Assignment 2 Due: Wednesday, Sept 30th 1. Determine the matrix of the linear operator L : R3 —> R3 with respect to the basis B and determine [L(:i:')],3 where B = {171,172,173}, L071) 2 171+2'172—i73, L072) = 2171—2172+173, L(173)= 172 + 363 and (5)3 = (1,2, —1). 2. 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) PerP(2,1,—2) b) reﬂuvzg) 3. Find the matrix of each linear operators with respect to the given basis B. a) L: P2 —) P2 defined by L(aa:2+b:c+c) = (a+b):z:2+c, B ={\$2+1,1L‘—1,:L‘2—£E+1}. b) D : P2 —> P2 deﬁned by D(aa:2+ba:+c) = 2ax+b, B = {3\$2+2m+1,x2—2x,\$2+x+1}. c) T : U —> U, where U is the Subspace of upper triangular matrices in M(2, 2), ewe :le antlers an; n l}- 4. Assume each of the following matrices is the matrix of some linear transformation with respect to the standard basis. Determine the matrix of the linear transformation with respect to the given basis B. a) [366 :3], B = {(1,s>,(1,2)} —2 6 4 b) —2 2 2 , B = {(1,0, 1), (1,1, —1), (1, —1,2)}. 3 —6 —3 5. Let V, W be ﬁnite dimensional vectors spaces over R. Give the formula for finding the matrix of a linear transformation L : V —+ M7 with respect to any basis B for V and any basis 0 for W. 6. Use your answer in 5. to ﬁnd the matrix of the following linear transformations with respect to the given bases. a) L : P2 —> R2 deﬁned by L(a.:1:2+b:1:+c) = (a+c,b—a), B = {x2+1,a:+1,\$2+a:—1}, (2+1) 0 b) T:1R2 -) M(2,2) deﬁned by T(a, b) = [ O a _ b], B = {(2, —1), (1,2)}, catttﬂtﬂttl 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., 31, S2, s3) and then construct any required matrices from these vectors (e.g., A = [31 52 53]). This will make it easier for you to work with the various elements Matrix Representation of a Linear Mapping Let L : R6 —» R6 be the linear mapping L[(a1,a2, a3,a4,a5, (16)] = (301— 0.2 + 0.3 + 70.4 + 20.5 — 306, a1 — a3 + a5 + Bag, (12 — 0.3 + 20,4 + (£5 + 505, (11 + 9112 + 9% — 7m; + 9%, —6a1 — a3 + 3a4 + 4% + 5125, a1+ 502 + 8% — 7m; + a5 — a6) (a) Find A, the matrix representation of the linear mapping L, with respect to the basis, 3, where B={ (17070)0v0:1))(1 071,071)0)1(_1)—1)0)0)—1)_1)) (0,0,—1,—1,0,0),(0,2,—2,2,—2,0),(1,2,0,1,2,0) } (b) Use A to ﬁnd L[(—1, —3,0,0, 5,4)1. [\D ...
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