NAME: DREXEL ID #: Sample Exam for Test 2 This is a sample test. It just gives you an idea about the formate of the exam. The actual exam will be a 60-minute test which is shorter than this sample test. No credit is given for answers like "Yes" or "No" without an explanation in your test. JUSTIFY ALL ANSWERS. 1. Let H = < 8 2 x3 3 ER3 : 2x1 +2 - 43 = 0 . : 4 (a) Prove that H is a subspace of R3. (b) Find a basis B for H and determine the dimension of H. 1 4 1 3 , is u in H? If yes, find the coordinates [u]B relative to the basis B you found in (b). (c) For the vector 2 2 2. Let V = R313, and let 8 b b-c W < 2 4 a b a b : c-a 0 a : a, b, c E R mi ; (a) Prove that W is a subspace of V. (b) Find a basis C for W and determine the dimension of W. 2 -1 (c) For the matrix A = 2 0 1 4 1 -1 -3 1 -1 , is A in W? If yes, find the coordinates [A]B relative to the basis C you found in (b). 3. Consider H L a b ER212 : 2a + b - c = 0 . c d (a) Prove that H is a subspace of R202. (b) Find a basis D for H and determine the dimension of H. (c) For the vector A = 2 r 2 -3 |is 7 in H? If yes, find the coordinates [A]D relative to the basis D you found in (b). 4. Let W={p(t) ER3(t): p(-1)+p(1) =0}. (a) Prove that W is a subspace of R3(t). (b) Find a basis B for W and determine the dimension of W. (c) Consider p(t) = t3 -t2 +3t +1. Is p(t) in W? If yes, find the coordinates [p(t)]B relative to the basis B you found in (b). 5. Let W = {p(t) E R3(t) : Ju p(t)dt = 0 }. 1
NAME: DREXEL ID #: (a) Prove that W is a subspace of R3(t). (b) Find a basis B for W and determine the dimension of W. (c) Consider p(t) = t3 - 3t2 + 2t +1. Is p(t) in W? If yes, find the coordinates [p(t)]B relative to the basis B you found in (b). 6. Let B = {3 + 2t, 1 + 3t2, -4t + 8t2 }. (a) Is B linearly independent? Why? (b) Is B a basis for R2(t)? Why? (c) If B is a basis for R2(t), find the coordinates [p(t)]B of p(t) = - 5 - 5t2 relative to B. 7. Let B 2 0 0 27 2 0 0 2 , , 1 1 1 1 22 '22 2 Is B a basis for R212? Why? - . 8. Let B 1 0 1 1