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Download CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the
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CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the vector space of real polynomials of degree at most two, and let L : P2 → P2 be the linear map defined by Lf = f 0 . Which of the following are eigenvalues of L? (a) −1 (b) 0 (c) 1 (d) 2 Solution: b) Explanation: The matrix of the {1, x, x2 } of P2 is 0 0 0 linear map with respect to the basis 1 0 0 2 . 0 0 The characteristic polynomial is therefore x3 , whose only root is 0. 2. Let A be a complex n × n-matrix and let λ be an eigenvalue of A. Which of the following are correct descriptions of the geometric multiplicity of λ? (a) the nullity of A − λI (b) the largest m such that (x − λ)m divides mA (x) (the minimal polynomial of A (c) the largest possible number of eigenvectors for A with eigenvalue λ in a linearly independent set (d) the largest m such that (x − λ)m divides pA (x) (the characteristic polynomial of A Solution: a), c) Explanation: a) is the definition. c) is the same as a) since the eigenvalues with eigenvalue λ are precisely the non-zero vectors in the null space of A − λI. b) is incorrect, since for example the geometric multiplicity of the eigenvalue 1 for the 2 × 2 identity matrix is 2, but the minimial polynomial is x − 1. d) is the definition of the algebraic multiplicity. 3. Let A be a complex n × n-matrix. Which of the following are equivalent to A being diagonalizable? (a) A has n distinct eigenvalues. (b) A satisfies its characteristic equation. (c) A is similar to a diagonal matrix. (d) Cn has a basis consisting of eigenvectors for A. Solution: c), d) Explanation: c) is the definition and we showed d) is equivalent. a) is sufficient but not necessary for A to be diagonalizable. b) holds for all square matrices. 4. Let A be a 4 × 4 complex matrix with minimal polynomial x3 + x2 . Which of the following could be the characteristic polynomial of A? (a) x4 + x2 (b) x4 + 2x3 + x2 (c) x4 + x3 (d) x3 + x2 Solution: b), c) Explanation: The characteristic polynomial has degree 4, and it has the same roots as the minimal polynomial, namely 0 and 1. Furthermore the characteristic polynomial must be divisible by the minimal polynomial, which is x2 (x + 1). The two remaining choices are possible since they have the form x3 (x + 1) and x2 (x + 1)2 . 5. Let v1 = √12 (1, 0, i), v2 = √13 (i, 1, 1) and v3 = √16 (i, −2, 1). Given that S = {v1 , v2 , v3 } is an orthonormal basis for C3 , determine which of the following appear in the coordinate vector for w = (0, 1, 1) with respect to S. (a) (b) √2 3 √1 2 (c) −1 (d) i Solution: a) Explanation: The coordinates are √ √ given by hw, vi i for i = 1, 2, 3, and are therefore −i/2, 2/ 3 and −1/ 6.