Math 271, Section 02, Spring 2017 Final Exam

Math 271, Section 02, Spring 2017 Final Exam, Friday, May 12, 2017 Instructions: Do all ten numbered problems. If you wish, you may also attempt the optional bonus problems. Show all work, including scratch work. Little or no credit may be awarded, even when your answer is correct, if you fail to follow instructions for a problem or fail to justify your answer. If you need more space, use the back of any page. If you have time, check your answers. WRITE LEGIBLY. NO DEVICES. 1. (15 points). Define T : R6 → R4 by T (~x) = A~x, where     1 −2 −1 0 2 1 1 −2 0 0 4 1 −1 2   3 1 1 2 . Take my word for it that B = 0 0 1 0 2 0 A= −2 4 0 0 0 1 −1 3 3 3 −5 7  1 −2 −1 −1 3 −2 0 0 0 0 0 0 is an echelon form of A. 1a. Find a basis for Ker(T ). 1b. Find a basis for Im(T ).   4 3 3 2. (20 points). Let B = 2 1 1. Compute B −1 . 1 0 1 3. (20 points). Let V, W, X be finite-dimensional vector spaces, let S : V → W and T : W → X be linear maps, and suppose that T S : V → X is the zero map. [That is, T S~v = ~0X for all ~v ∈ V .] 3a. Prove that Im(S) ⊆ Ker(T ). 3b. Prove that rank(S) + rank(T ) ≤ dim(W ). 4. (25 points). Define T : P2 (R) → P2 (R) by T (p(x)) = p(x) + x · p(2). For example, T (x2 + 1) = (x2 + 1) + x · (5) = x2 + 5x + 1. Take my word for it that T is linear. 4a. Let α = {1, x, x2 }. Find the matrix [T ]αα representing T with respect to α. 4b. Verify that T is diagonalizable by finding a basis of P2 (R) consisting of eigenvectors of T , and writing down the associated diagonal matrix. 5. (20 points). Let V, W be vector spaces, let T : V → W be a linear transformation, and let α = {~v1 , . . . , ~vn } ⊆ V . Define β = {T~v1 , . . . , T~vn }. Suppose α is a basis for V , and that β is linearly independent. Prove that T is one-to-one. 6. (15 points). Answer the following questions and provide brief explanations. 6a. Let A be a 7 × 12 matrix such that for every ~b ∈ R7 , the equation A~x = ~b has at least one solution ~x ∈ R12 . What is the nullity of A? [That is, what is dim(Ker(A))?] 6b. Let B be a 3 × 3 matrix such that det(B) = −3. Can there be a vector ~b ∈ R3 such that the equation B~x = ~b has no solution? 7. (20 points). In each part of this problem, a different square matrix will be described. For each, answer: • Yes, if the matrix is definitely diagonalizable, • No, if the matrix is definitely not diagonalizable, or • More Info, if more information is needed to determine whether or not the matrix is diagonalizable. Don’t forget to justify your answers. 7a. A ∈ M4×4 (R) has characteristic polynomial t(t + 5)3 . 7b. B ∈ M3×3 (R) has determinant 0, trace 5, and each column summing to 7. 8. (20 points). Find an orthonormal basis for  1  A = −1 1 R3 consisting of eigenvectors of the matrix  −1 1 1 −1 . −1 1 9. (25 points). Solve the system of differential equations x0 (t) = 2x(t) + 3y(t) y 0 (t) = 2x(t) + y(t) subject to the initial condition x(0) = 5, y(0) = 10. 10. (20 points). Let T : R7 → R7 be a linear map with the property that T (T (~v )) − 2T (~v ) = 3~v for all ~v ∈ R7 . Prove that the only possible real eigenvalues of T are −1 and 3. OPTIONAL BONUS A. (2 points). Find a 2 × 2 matrix A ∈ M2×2 (R) such that A is not diagonalizable, but A2 is diagonalizable. (Don’t forget to verify your claims.) OPTIONAL BONUS B. (2 points). Let T : Rn → Rn be a linear map with adjoint T ∗ : Rn → Rn . Suppose that T ∗ T = O, where O is the zero map O : Rn → Rn . Prove that T = O.