2017-08-18 23-43.pdf - UH Math 4377/8308 Dr Heist Fall 2013...

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Unformatted text preview: UH - Math 4377/8308 - Dr. Heist - Fall 2013 HW 3 Due 09/19, at the beginning of class. Use regular sheets of paper, stapled together. Don’t forget to write your name On page 1. 1. Determine if the following subsets of R5 are subspaces. (a) (0.5 points) {(01.02.03) 6 R3 : 201 — 3% = 0} (b) (0.5 points) {(al,a2,o3) e R3 :51 — 203 + a3 = 1} (c) (0.5 points) {(a1,a2,ag) E R3 : 201 =2 as} (d) (0.5 points) {(a1,aa,as) E R3 : 261 = 5413 and 40.2 = a1 + 0.3} l tannins if the following subsets of the vector space of 2 x 2 matrices with real entries are subspaces. on may assume as true that the set of 2 x 2 matrices with real entries forms a vector space with the usual addition and scalar multiplication. (a) (1 point) {(2; C62) :a1,a2,a3 E R} (b) (1 point) {(3 Z?) :al,az,a3 6 JR} .(1 point) A real-valued flmction f defined on the real line is called an even function if fit) = f(—t) for each real number t. Prove that the set of even functions is a subspace with the usual addition and scalar multiplication for functions. You may assume as true that the set of real-valued functions 3‘ defined on the real line is a. vector space with the usual addition and scalar multiplication for functions. 4. (1 point) Let W1, W2 be two subspaces of a vector space V. Prove that the intersection W1 fl W2 is also a subspace of V. 5. (1 point) Section 1.3, Problem 18. ,(2 points) Let V be the vector space of all functions f : IR —) 1R. Let W1 2 {f : R -——> RUG) = 0}. ’rove that W1 is a subspace of V. Find a. subspace W2 C V such that V = W1 GB W2. Prove all your statements. (This is the problem I posed in class, and I couldn’t resist putting it on the homework.) 7. (1 point) Section 1.3, Problem 28 (Work with F = IR only. This allows you to disregard the half— sentence “Now assume that F is not of characteristic 2 (see Appendix C),”.) .(1 extra point) Let W1 , W2 be two ‘subspaces of a vector space V. Prove that the union W; U W2 is a subspace of V if and only if W2 _C_ W1 or W1 C_Z W2. UH — Math 4377/6308 — Dr. Heier - Fall 2011 HW 3 Due 09/14, at the beginning of class. Use regular sheets of paper, stapled together. Don’t forget to write your name on page 1. 1. Determine if the following subsets of R3 are subspaces. (a) (0.5 points) “aha-baa) E R3 : 201 - 303 = 0} (b) (0.5 points) {(a1,aa.a3)e R3 : a1 — 20; +0.3 = 1} (C) (0-5 PointS) {(a1,ag,a3) E R3 = 2a: = as} (d) (0.5 points) {(a;,a2,a3) e R3 : 2:11 = 5:13 and 40.2 = a1 + a3} @Determine if the following subsets of the vector space of 2 x 2 matrices with real entries are subspaces. You may assume as true that the set of 2 x 2 matrices with real entries forms a vector space with the usual addition and scalar multiplication. (a) (1 point) { (2; ‘39) :a1,a2,a3 e R} (b) (1 Point) ((2: :32) : 01,42,423 6 JR} (1 point) A real-valued function f defined on the real line is called an even function if f (t) = f (—t) for each real number t. Prove that the set of even functions is a subspace with the usual addition and scalar multiplication for functions. You may assume as true that the set of real-valued functions f defined on the real line is a. vector space with the usual addition and scalar multiplication for functions. 6 (1 point) Let WI, W2 be two subspaces of a. vector space V. Prove that the intersection W1 n W2 is .- o a subspace of V. 5. (1 point) Section 1.3, Problem 18. .Let W1 = ((a1,a2,a1 +a2)la1,a2 E R} C R3. (a) (1 point) Give an example of a subspace W2 such that W1 EB W2 = R3. Justify your answer. (b) (1 point) Let W2 = {(01,01 + a2,a2)lal,a2 E R} C R3. 13 W1 69 W2 = R3? 13 W1 + W2 2 R3? Justify your answer. 7. (1 point) Section 1.3, Problem 28 (Work with F = R only. This allows you to disregard the half- sentence “Now assume that F is not of characteristic 2 (see Appendix C),”.) (1 extra point) Let W1, W2 be two subspaces of a vector space V. Prove that the union W1 U W2 is a. subspace of V if and only if W2 9 W1 or W1 g W2. ...
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