MATH 4140/7140 Homework 2 Due 6/27/2016 Do all problems on different paper. I will not grade anything written on this page. Show your work. Answers without work will be given little or no credit. The zero matrix or zero vector will be denoted 0. 1. Find the inverse of each of the following matrices. \u0012 \u0013 1 1 1 2 0 0 2 4 A= , B = 0 1 1 , C = 0 3 0 , 1 1 0 0 1 0 0 5 1 3 3 6 1 D= 2 3 8 3 2. If A is row equivalent to B and B is row equivalent to C, show that A is row equivalent to C. This shows, in particular, that any two nonsingular matrices are row equivalent by replacing B with the identity matrix. 3. (a) Give examples of nonsingular 2 2 matrices A and B such that A + B is singular. (b) Give examples of nonsingular 22 matrices A and B such that A+B is nonsingular, but (A + B)1 6= A1 + B 1 . 4. (a) Calculate the determinant of each of the following matrices. \u0012 \u0013 2 0 0 3 1 2 2 1 5 A= , B= 2 3 0 C= 2 4 4 2 0 1 1 5 5 4 (b) Use your answers from (a) to calculate each of the following. (i) (ii) (iii) (iv) det 5A det BCD det D1 det BDT 5. Use Cramer's rule to solve the following system of equations. 2x1 + x2 3x3 = 0 4x1 + 5x2 + x3 = 8 2x1 x2 + 4x3 = 2 1 2 1 3 D = 1 2 2 1 4 0 6. (a) Let S denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by {an } = {an }, {an } + {bn } = {an + bn }. Show that S is a vector space by showing that it satisfies the axioms. (b) Let T be the subset of S consisting of all infinite sequences with all but finitely many terms equal to zero. Show that T is a subspace of S. 7. In each of the following, for the given vector space, determine if the given set is (i) a spanning set, and (ii) linearly independent. (a) The span of {(2, 1, 2)T , (3, 2, 2)T , (1, 0, 1)T } in R3 . (b) The span of {(1, 2)T , (2, 1)T , (3, 1)T } in R2 . 8. In each of the following, determine if the given set is a subspace of the stated vector space. (a) {(x1 , x2 )T | x1 = 3x2 } in R2 . (b) The set of all 2 2 triangular matrices in R22 . (c) The set of all polynomials of degree 5 in P6 . 9. Calculate the null space of \u0012 A= 2 4 0 1 3 2 2 2 \u0013 10. Let U and V be subspaces of a vector space W . (a) Prove that U V is a subspace of W . (b) Find choices for U, V, and W to show that U V need not be a subspace of W . Explain why your counterexample works. 2