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Exam # 1 Spring 2005 MATH 2270-01 Instructor: Oana Veliche Time: 50 minutes NAME: ID#: INSTRUCTIONS (1) Fill in your name and your student ID number. (2) Justify all you answers. Correct answers with no justification will not be given any credit. (3) No books, notes or calculators may be used. Page # 2 3 4 5 6 7 8 Total Max. # points 16 10 12 9 18 20 15 100 # Points 1 2 Problem 1. Consider the following matrix equation 1 0 1 1 2 2 0 1 1 A~x = ~b: x 0 y = 3 . z 1 (8 points) (a) Find the solution to this system using elementary row operations. (8 points) (b) Find the inverse of the matrix A. 3 (4 points) (c) Check that the matrix obtained in (b) is indeed the inverse of A. (6 points) (d) For invertible matrices A, there is a unique solution to A~x = ~b, and a formula for this solution which uses inverse matrix. Use this formula to resolve the system in part (a). 4 (6 points) Problem 2. Explain (in general) when an equation A~x = ~b, with A a n × m matrix, has a unique solution? (6 points) Problem 3. Suppose that T : R3 columns of the matrix A from Problem 1: 1 0 , T T 1 = 1 0 0 Find T 3 . 1 → R2 is a linear map and that we know what T does to the 1 0 −1 1 2 2 = . = , and T 2 2 1 1 5 Problem 4. We consider a linear transformation T (~x) = A~x for the following matrix A with its reduced row-echelon form: 1 3 2 −1 −1 1 3 0 0 −1 0 3 and rref (A) = 0 0 1 0 1 A= 2 6 5 0 0 0 1 2 0 0 0 1 2 (5 points) (a) Find a basis for the image of T . (4 points) (b)Express the fifth column of A as a linear combination of the basis vectors you found in part (a). 6 (8 points) (c) Find a basis for the kernel of T . Verify that your set of vectors is linearly independent. (10 points) Problem 5. Prove that any linear transformation T : R2 → R2 , that is given by a rotation has the form T (~x) = A~x, where a −b , a2 + b2 = 1. A= b a 7 Problem 6. (4 points) (a) Give the definition of linearly independent vectors. (6 points) (b) Give the definition of a subspace of Rn , and give an example of a subspace. Problem 7. True or False ? (Justify all your answers.) (5 points) (a) If A and B are two matrices of the same size, then the identity holds: (A − B)(A + B) = A2 − B 2 . (5 points) (b) There exist a 4 × 2 matrix A and a 2 × 5 matrix B such that dim(Ker(AB)) = 2. 8 (5 points) (c) The B-matrix of the linear transformation T : R2 → R2 , given by the reflection with respect to a line L, that goes through (0, 0), is the same for all the lines L. (5 points) (d) If W is a subspace of Rn of dimension m, any linearly independent vectors of length m form a basis for W . (5 points) (e) If A and B are invertible matrices, then AB is similar to BA.