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240la - Math 240 Some More Challenging Linear Algebra...

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Math 240: Some More Challenging Linear Algebra Problems Although problems are categorized by topics, this should not be taken very seriously since many problems fit equally well in several different topics. Note that for lack of time some of the material used here might not be covered in Math 240 . Basic Definitions 1. Which of the following sets are linear spaces? a) { X = ( x 1 ,x 2 ,x 3 ) in R 3 with the property x 1 2 x 3 = 0 } b) The set of solutions x of Ax = 0, where A is an m × n matrix. c) The set of 2 × 2 matrices A with det( A ) = 0. d) The set of polynomials p ( x ) with integraltext 1 1 p ( x ) dx = 0. e) The set of solutions y = y ( t ) of y ′′ + 4 y + y = 0. 2. Which of the following sets of vectors are bases for R 2 ? a). { (0 , 1) , (1 , 1) } b). { (1 , 0) , (0 , 1) , (1 , 1) } c). { (1 , 0) , ( 1 , 0 } d). { (1 , 1) , (1 , 1) } e). { ((1 , 1) , (2 , 2) } f). { (1 , 2) } 3. For which real numbers x do the vectors: ( x, 1 , 1 , 1), (1 ,x, 1 , 1), (1 , 1 ,x, 1), (1 , 1 , 1 ,x ) not form a basis of R 4 ? For each of the values of x that you find, what is the dimension of the subspace of R 4 that they span? 4. If A is a 5 × 5 matrix with det A = 1, compute det( 2 A ). 5. Let A be an n × n matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix A is invertible”? a) The columns of A are linearly independent. b) The columns of A span R n . c) The rows of A are linearly independent. d) The kernel of A is 0. e) The only solution of the homogeneous equations Ax = 0 is x = 0. f) The linear transformation T A : R n R n defined by A is 1-1. g) The linear transformation T A : R n R n defined by A is onto. h) The rank of A is n . 1
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i) The adjoint, A , is invertible. j) det A negationslash = 0. Linear Equations 6. Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y . Give a proof or counterexample for each of the following. a) If n = k there is always at most one solution. b) If n>k you can always solve AX = Y . c) If n>k the nullspace of A has dimension greater than zero. d) If n<k then for some Y there is no solution of AX = Y . e) If n<k the only solution of AX = 0 is X = 0. 7. Let A : R n R k be a linear map. Show that the following are equivalent. a) A is 1-to-1 (hence n k ). b) dim ker( A ) = 0. c) A has a left inverse B , so BA = I . d) The columns of A are linearly independent. 8. Let A : R n R k be a linear map. Show that the following are equivalent. a) A is onto (hence n k ). b) dim im( A ) = k . c) A has a right inverse B , so AB = I . d) The columns of A span R k . 9. Let A be a 4 × 4 matrix with determinant 7. Give a proof or counterexample for each of the following. a) For some vector b the equation A x = b has exactly one solution. b) some vector b the equation A x = b has infinitely many solutions. c) For some vector b the equation A x = b has no solution. d) For all vectors b the equation A x = b has at least one solution. 10. Let A and B be n × n matrices with AB = 0. Give a proof or counterexample for each of the following.
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