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Math 1502 Practice Final Exam 1a. Find the eigenvalues and corresponding eigenvectors to the matrix A= 1 6 64 Find A15 (Show all work). Write the...

I need help on 1c,1d, and 1e.

Math 1502 Practice Final Exam 1a. Find the eigenvalues and corresponding eigenvectors to the matrix A = ± - 1 6 6 4 ² b. Find A 15 (Show all work). c. Write the quadratic form Q associated with A and plot the curve Q x ) = 2. d. Determine whether Q x ) = ¯ x T A ¯ x is positive definite, negative definite, or indefinite. e. Show that if A is symmetric then the eigenvectors associated with different eigenvalues are orthogonal. f. Find det( A ) where A = 1 2 2 1 1 0 1 1 1 1 0 2 2 1 1 0 . 2.a Show whether or not the set of vectors V = { ¯ x = t 1 2 1 + s 1 1 1 + 1 0 2 , -∞ < t, s < ∞} is a subspace. b. Given the vectors ¯ x 1 = 1 1 - 2 0 ¯ x 2 = 1 0 - 1 1 ¯ x 3 = 1 0 1 1 , find an orthonormal basis for V = Span(¯ x 1 , ¯ x 2 , ¯ x 3 ). c. Find the orthogonal projection of ¯ x = 1 1 1 4 in V from part b. 3. (a) Find a basis for col A and Nul A of the following matrix. A = 1 1 - 1 0 - 1 1 0 1 1 0 1 1 - 1 1 - 2 1 0 1 1 0 (b) Find the dim(Nul A ) and rank A (c) Find a basis for the Nul A T and col A T .
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4a. Find and graph all the solutions to the equations A ¯ x = ¯ b where A = ± 1 2 1 0 1 1 ² , and ¯ b = ± 1 - 1 ² b Amoung the solutions to the part a. find the one that is closest to the origin. What subspace is it perpendicular to? c. Find the least squares solution to A ¯ x = ¯ b , where, A = 2 2 0 2 0 1 , and ¯ b = 1 2 1 d. Find the orthogonal complement to the subspace V = span(¯ x 1 , ¯ x 2 ) where ¯ x 1 = [1 , 2 , 1 , 1] T , ¯ x 2 = [1 , - 1 , 0 , - 1] T . 5. Show whether the following series converge absolutely, converge conditionally, or di- verge. (a) X k =2 ( - 1) k k 3 + 2 k k 5 - k 2 + 1 (b) X j =1 ( - 1) k j + 1 ( j + 5) 2 6. (a) Find the power series representation for Z x 0 1 - e t t dt . (b) Find the radius of convergence and interval of convergence of the series X j =2 ( - 3 / 4) j j + 1 ( x + 1) j . 7. (a) Find P 5 for sin 3 x . (b) With f ( x ) = e 2 x find an upper for R n for x [0 , 2] (c) Find the Taylor series for (1 + 5 x ) - 2 / 3 at x = 0 and determine its radius of convergence. 8. Determine if the improper integrals are convergent or not. (Show work.) (a) Z 2 1 1 x 2 - 1 dx (b) Z 1 ( x + 1) - 1 / 2 dx . 9. Let A be an n × n matrix. a Show that if μ is an eigenvalue of A then μ k is an eigenvalue of A k . b Show that if A has n distinct eigenvalues then A is diagonalizable. c Find A - 1 where A = 2 1 0 1 1 1 2 2 1 .
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