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Unformatted text preview: Math 136 Assignment 6 Due: Wednesday, Mar 3rd 1. Determine, with proof, which of the following are subspaces of the given vector space.
Find a basis for each subspace. a) A = {(1:1,$2,a;3) E R3 l 2:121 +233 =0,:1:1+a:2—w3 =0} ofR3.
b) B: {ax2+b:r+c€ P2b2—4a.c=0} ong.
c) C={ax2+bzr+ce P2a.—c=b} ong. d) D = {[i 2] e M(2, 2)  ad — bc = 0} of M(2, 2). e) E = {[21 2] e M(2,2) a — b+c= 0} of M(2,2). 2. Determine, with proof, whether the given set is a basis for P2.
a) {1 +5w+m2,2+:r,3+21'+x2)}.
1)) {1+ 23: + 32:2,3 + 2$+$2,5 + 63: + 7x2}. 3. Invent a basis for each of the following subspaces of 1V! (2, 2).
a) The set of all 2 X 2 lower triangular matrices. b) The set of all 2 x 2 diagonal matrices. 4. Let S = {171, . . . 47"} be a set of vectors in a vector space V Prove that spanS’ is a
subspace of V 5. Let S = {(a.,b) E R2  I) > 0} and deﬁne addition by (a, b) + (c, (1) = (011+ bc,bd) and
deﬁne scalar multiplication by k(a, b) = (kabk'l, bk). Prove that S is a vector space over 1R.
6. Let V and W be vector spaces and let T : V —) W be a linear mapping. a) Suppose that {171,...,17k} a linearly dependent set in V Prove that {T(17,), . . . , T(17k)}
is a linearly dependent set in W. b) Suppose {271, . . . ,17,"} is a set of vectors in V and that {TU1’1), . . . ,T(17,,,)} is
a spanning set for IV. Prove that the range of T is W". 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., $1, 52, s3) and then construct any required matrices from these vectors
(e. g., A = [s1 s2 33]). This will make it easier for you to work with the various elements. (a) Determine if p(:c) = —2 — 3.r — 22:2 + 7173 belongs to the span of S = {p1,p2,p3,p4,p5}, where
p1(;r) = 1 + (I: — r02 — 1‘3
192(1) 2 —3 — a: + 22:2 + 4x3 — $4
[)3(.’L‘) = 5:: + x2 — :z:4
174(33) = —1 + 3:33
p5(rc) = 2 — 1:2 + 23:4 (b) Determine if the following subset of R5 is linearly independent:
{(0, 8, 0, —8,0), (—2,0, 1, —1,3), (5, —4, 1,3, 2), (—3, 7, 0, —1, —1), (0,0, 0, 3, 1)}. q. . 7‘ _ 123 —28 4 41 0
(c) DetermineifthesetS—{[4 5 6]’[ 6 9 _1]a[_2 0 _3]: 8 —3 2 14 —5 —4 12 10 12 W ‘
[0 1 10 ] i [ _12 _9 —8 ] , [ 12 20 18] } forms a. basis for M(2,3). ...
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 Spring '08
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 Linear Algebra, Algebra

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