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assign7 - Math 136 Assignment 7 Due Wednesday Mar 9th 1...

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Unformatted text preview: Math 136 Assignment 7 Due: Wednesday, Mar 9th 1. Prove that each of the following sets is a subspace of the given vector space. Find a basis for the subspace and hence determine the dimension of the subspace. a)81:{a$2+bx+c]a+b+c=0} ofP4 a3 a4 1 0 . . 4 . . 2 2 2. Find a ba81s for R containing the vectors _1 , 1 3 1 3. Leth{1+2a:+:c2,1+m,1~33—2232,1+333—3332,1+4m—42:2}. a) Prove that B is not a basis for P2. b) Find a subset of B that is a basis for P2. 4. Let B = { B] , B] } and C = { E] , [fl } and let L : R2 --> R2 be the linear mapping such that mg 2 [L(:Z’)]C a) Finnggp. a1 a2 5. Let'll‘={[O a 3 J l (11, a2, a3 6 R} be the vector space of upper triangular matrices with 0 O 0 0 0 1 and B = { [3 O] , [_1 3] , [1 “3:” are both bases for T. Find the change of coordi- standard addition and scalar multiplication of matrices. The sets 8 : { [1 0] , [0 1] , [0 0] } 0 3 0 0 0 1 nates matrix from B to S. 6. Let S = {(a,b) E R2 l b > 0} and define addition by (a,b) @ (0,61) = (ad+ bc,bd) and define scalar multiplication by k (D (a, b) = (kabk’1,bk). Prove that S is a vector space over R. 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. Vector Spaces Make use of the MATLAB commands rref and rank to help you solve the problems related to vector spaces below. If you’re not familiar with any of these commands, look them up in MATLAB’S Help section. For questions that involve a set of vectors, enter each vector separately, giving it a name (e. g, 31, s2, 53) and then construct any required matrices from these vectors (eg, A = [$1 $2 83]). This will make it easier for you to work with the various elements. (a) Determine if 13(30) 2 4 — 32: + 63:2 — 2x3 — 59:4 belongs to the span of S = {p1,p2,p3,p4,p5}, where 191(93) : 3 + :27 — 2:4 p2($) 4 + 53: — 59:2 + 5503 + 21r4 193(33) : 2:2 + $3 — $4 p4($) : —5 — 3$ + 12:2 + 6563 + 6:124 195(23) = 2 — 23:2 + 2:174 (b) Determine if the following subset of R5 is linearly independent: {(—7,9,9,0,6), (1,2, ~3, ~4, ~5), (0,0,0, —1, —4), (1, 1, 1, 1, 1), (5,5, —2,3, —6)}. (C)DetermineifthesetS:{[—3 17],[1 01],[’2 5 *1], 7 ~5 7 O —1 O 7 ~13 9 1 3 5 7 1 —6 4 4 -5 . ' [2 4 6]’[—2 _9 —4]’[O _1 2]}formsabasisfor/\/l(2,3). ...
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