assign6 - Math 136 Assignment 6 Due Wednesday Mar 3rd 1...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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,$2,a;3) E R3 l 2:121 +233 =0,:1:1+a:2—w3 =0} ofR3. b) B: {ax2+b:r+c€ P2|b2—4a.c=0} ong. c) C={ax2+bzr+ce P2|a.—c=b} ong. d) D = {[i 2] e M(2, 2) | ad — bc = 0} of M(2, 2). e) E = {[21 2] e M(2,2) a — b+c= 0} of M(2,2). 2. Determine, with proof, whether the given set is a basis for P2. a) {1 +5w+m2,2+:r,3+21'+x2)}. 1)) {1+ 23: + 32:2,3 + 2$+$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 {TU-1’1), . . . ,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). ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern