Practice Exam #1 Math 2270, Spring 2005 Problem 1. Consider the following matrix equation Ax = b: 2 2 1 x 2 0 1 −1 y = 3 . 1 1 1 z 0 (a) Find the solution to this system using elementary row operations. (b) Find the inverse of the matrix A. (c) For invertible matrices A, there is a unique solution to Ax = b, and a formula for this solution which uses inverse matrix. Use this formula to resolve the system in part (a). (You should get the same answer.) (d) Explain (in general) why the formula you used in part (c) gives a correct solution, and why this is the only solution. Problem 2. Suppose that T : R3 → R2 is a linear map and that we know what T does to the columns of the ,matrix A from Problem 1: 2 1 2 −1 0 1 1 −1 0 , T , and T . T = = = 1 1 1 1 1 1 2 3 . (Hint: Use the abstract definition of what a linear function is, together with Find T 0 your answer to Problem 1, part (a).) Problem 3. We consider a linear transformation T (x) = Ax, for the following matrix A with its reduced row-echelon form: 1 0 0 1 1 2 1 −1 3 0 1 −1 0 −4 2 −1 1 and rref (A) = 0 1 0 A= 0 0 1 −1 −2 0 3 −5 5 −2 0 0 0 0 0 1 0 0 1 1 (a) Find a basis for the image of T . (b)Express the fifth column of A as a linear combination of the basis vectors you found in part (a). (c) Find a basis for the kernel of T . Verify that your set of vectors is linearly independent. (d) State the Rank-Nullity theorem and verify that this theorem hold for the matrix A above. Problem 4. Prove that any linear transformation T : R2 → R2 , that is given by the projection of a vector on a line L has the form T (x) = Ax, where u21 u1 u2 , u21 + u22 = 1. A= u1 u2 u22 Problem 5. (a) Give the definition of the kernel of a matrix n × m. (b) Give the definition of basis of a subspace of Rn . Problem 6. True or False ? (Justify all your answers.) (a) If a square matrix A satisfies A2 = A, then A is the 0 matrix or the identity matrix. (b) If A is any matrix then the image of A and the image if rref (A) are the same. (c) If the matrix A is similar to the matrix B, and A is invertible, then B must be invertible as well. (d) There are matrices 3 × 3, A for which im(A) = ker(A). (e) If T : Rn → Rm is a linear transformation, then any linearly independent vectors are transformed by into linearly independent vectors.