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Sample Midterm 1 (Minor 2012) Sample Midterm 2 (2010) Sample Midterm 3 Other Midterm Problems Review: Computational problems Problems 1 – 5 Let A a1 a 7 . A I 55 1 1 1 0 1 2 0 0 1 0 2 0 0 1 1 0 3 2 0 3 1 9 1 23 3 I 55 ~ 0 0 5 2 16 1 36 5 0 0 0 0 12 5 39 0 0 1 79 11 1. Find the dimension of each of the following spaces. a) Row ( A) b) Col ( A) c) Nul ( A) 0 1 0 0 0 2 3 0 0 0 0 0 1 0 0 4 5 6 0 0 0 1 1 0 0 2 2 1 0 0 1 5 4 1 0 0 4 3 1 1 0 17 13 1 0 2. Verify the following relationship numerically. a) rk ( A) dim[ Nul ( A)] n 3. a) b) c) Find a basis for each of the following spaces. Row ( A) Col ( A) Nul ( A) 4. Express a 4 , a6 , and a 7 as a linear combination of the other column vectors. 5. Does the system Ax b have a solution for every b? Give a proof or a counterexample. 6. How can you tell that det( A) 0 quickly? 1 2 3 4 5 6 7 8 a) A 9 10 11 12 0 0 0 0 1 2 0 3 6 5 0 4 b) A 7 8 0 9 12 11 0 10 1 2 3 4 5 6 7 8 c) A 9 10 11 12 10 20 30 40 0 0 0 0 1 1 2 3 7. Given A 4 6 7 and det( A) 2 , find det( B ) . 5 8 9 1 2 41 3 1 0 0 42 0 d) B 4 a) B 4 6 43 7 15 5 8 99 9 1 5 8 9 e) B 2 b) B 4 6 7 3 1 2 3 f) BA I 10 1 2 3 g) B 40 c) B 4 6 7 50 50 80 90 3 6 7 28 39 4 5 6 8 7 9 2 2 3 2 1 3 6 7 6 4 7 8 9 8 5 9 8. Given the equations below, find det( A) . 0 8 1 1 0 4 a) A4 0 , A3 0 , A0 0 4 10 5 2 2 5 1 1 0 4 0 3 b) A4 0 , A3 0 , A0 5 5 2 2 5 4 3 9. Let A be the following matrix. 0 2 1 0 0 1 0 0 A 3 2 3 1 5 1 1 a) For what values of is A invertible? b) Assuming A is singular, find the rk ( A) , dim[ Nul ( A)] , and the Nul ( A) . 10. Let A be a 4 4 matrix of all ones. 1 1 1 1 1 1 1 1 A 1 1 1 1 1 1 1 1 a) Show A2 4 A . b) Let B A 2 I . Show 8B B 2 12 I 0 c) Find B 1 . 11. Let A be the below matrix. 1 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 A ~ 2 0 23 0 0 0 43 2 0 2 1 1 1 1 0 0 a) Verify that A is invertible. b) Suppose b1, b2, b3, and b4 are real numbers. Show that there is exactly one P3 polynomial such that the following equations are true. p(1) b1 , p(0) b2 , 1 1 p( x)dx b3 , and p(1) b4 Review: Conceptual questions 1. A is an m by n matrix. Justify your answer with a proof if the statement is true, or give a counter-example or proof if the statement is false. a) If Ax b is not consistent, then rk ( A) m . b) If Ax b is consistent and n m , then there are infinitely many solutions. c) If Ax b is consistent and n m , then there is exactly one solution. d) If Ax b is consistent and rk ( A) m , then there is exactly one solution. e) If Ax b is consistent and rk ( A) n , then there are infinitely many solutions. f) If rk ( A) n , then Ax b is consistent for every b. 2. a) b) c) d) A is an m by n matrix of rank r. What is the relationship between m, n, and r in each case? A has an inverse. Ax b has a unique solution for every b in Rm. Ax b has a unique solution for some, but not all b in Rm. Ax b has infinitely many solution for every b in Rm. Review: Proofs 1. Suppose S {v1 , , v n } is a basis of R n , and A is an n n invertible matrix. a) Prove the following set B { Av1 , , Av k } is independent when k n . b) Prove the following set C { Av1 ,, Av n } is a basis for R n . 2. Let A span{u, v} and let B span{u, v, u v} . Prove A B . 3. Suppose A is m n . If Col ( A) Row( A) , prove m n . 4. Suppose A 2 0 . Prove A is not invertible.