MA 405 Test 4 Review

# MA 405 Test 4 Review - Test 4 Review The test covers...

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Test 4 Review The test covers sections 4.1 (Kernel and Range only), 4.3, 6.1, 6.3, and the topics covered in class from 5.1, 5.2, 5.5. Know all theorems and deﬁnitions presented in class. Know how to work all homework problems, especially those not collected for grading. Deﬁnitions : Kernel, range, similar matrices, eigenvalue, eigenvector, characteristic polynomial (det( λI - A )), diagonalizable, matrix exponential, length, distance, angle, dot product (standard inner product), orthogonal, projection, orthogonal complement, orthogonal set, orthonormal set Theorems : 4.3.1, 6.1.1, 6.3.1, 6.3.2, 5.1.1 (true for R n ), 5.5.1 and 5.5.2 (where V = R n ) Sample Problems : 1. Range and Kernel, see test 3 review 2. Let T : R 2 R 2 be the linear transformation deﬁned by T ( x 1 , x 2 ) T = ( - x 2 , 2 x 1 ) T . Consider the two bases A = { v 1 = (1 , 1) T , v 2 = (0 , 1) T } and B = { w 1 = (1 , 0) T , w 2 = (2 , 1) T } for R 2 . Find (a) The matrix A, representing T with respect to A . (b) The matrix B, representing T with respect to B . (c) The matrix P, such that B = P - 1 AP . 3. Consider the matrix A = 4 - 5 1 1 0 - 1 0 1 - 1 (a) Find the eigenvalues and corresponding eigenspaces for A . (b) Find a matrix P such that P - 1 AP is diagonal. (c) Find e A . 4. For what values of α will the following matrix fail to be diagonalizable? 4 6 - 2 - 1 - 1 1 0 0 α 5. page 341 #12 Show that a nonzero nilpotent matrix is defective. Hint: Find the eigen- values of a nilpotent matrix. 1

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6. If A 2 = αA for some scalor α ± = 0, show that the eigenvalues of A must all be 0 or α . 7. Reminder: Similar matrices have the same eigenvalues because they have the same characteristic polynomial. Do they necessarily have the same eigenvectors? Prove or ﬁnd a counterexample. Hint: Consider a 2 × 2 diagonalizable matrix to get an idea. 8. (a) Show that a 2 × 2 symmetric matrix must have real eigenvalues. Hint: For a general 2 × 2 symmetric matrix ﬁnd the characteristic polynomial and compute the discriminant. (b) Show that a 2 × 2 skew-symmetric matrix must have pure imaginary eigenvalues. That is all eigenvalues are a multiple of
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MA 405 Test 4 Review - Test 4 Review The test covers...

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