MATH 2510A Final 0910

MATH 2510A Final 0910 - 1¥~§ (£- 3 E) Pagelof 3 : m v 7s...

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Unformatted text preview: 1¥~§ (£- 3 E) Pagelof 3 : m v 7s a: 42 SC in e The Chinese University of Hong Kong f—gifl, i ;$*—$ $E§j§¥flfifi #ti Course ExaminationflTerm, 2009 - 2010 77st 5 5&i7f.&z% fi Course Code & Title ._ MAT2310A LINEAR ALGEBRA AND APPLICATIONS 3% Fa'i Time allowed ‘31: itfiaifi Student ID. No. SHOW ALL NECESSARY WORK TO GET CREDIT FOR SOLUTIONS (NOTE: THERE ARE 100 POINTS TOTAL ON THIS EXAMINATION) 1. Given that 1 1 —2 1 4—8 -6 3~1 2 0 (a) (2 points) Find the reduced row echelon form of A. (b) (6 points) Find a basis for the null space of A and the nullity of A. (c) (6 points) Find a basis for the column space of A and the column rank of A. 2. (10 points) Given that rank A = 3, and Find the values of a and ,6, if any. 3. Let A = {3.1, a2,a3} and B = {b1,b2, b3} be ordered bases for 732, Where a1=t2+t+1, a2=t2+2t+3, a3=t2+1 b1=t+1, b22152, b3=t2+1. h=2t2—6. (a) (5 points) Find the coordinate vector of h with respect to B. (b) (6 points) Compute the transition matrix from the B—basis to the A—basis. (c) (2 points) Find the coordinate vector of h With respect to B using the transition matrix from the A—basis to the B—basis. 'Course Code fl 3 $5215 I .... ................................................... .. g 2 H0”? 3 E) Pag620f3 4. Given that F4X + E4X = 2F2E2X + w. (a) (5 points) Calculate E = uvT and F = vTu. (b) (5 points) Solve for X , if any. 5. (a) (6 points) Apply the Gram—Schmidt process to (b) Let 1 2 1 1 A = 1 3 ‘2 and c = 5 0 ——1 0 0 O 0 1 2 i. (6 points) Find the least squares solution to the system Ax = c. ii. (2 points) Use 5(b)i, compute the squared error. iii. (2 points) Assume that a matrix P is a projection matrix. Show that P is symmetric and P2 = P. 1 2 O A: 0 2 0 -—2 —1 —l (a) (4 points) Find the characteristic polynomial of A. (b) (12 points) Find all eigenvalues and corresponding eigenvectors of A? (c) (2 points) Why is A diagonalizable? (d) (4 points) Find an invertible matrix K and a diagonal matrix D such that K ‘IAK = D. " Course Code 2?} E 353% I .... ..................................................... ._ g 3 ED}? 3 E) Page3of3 7. Answer each question separately. Show work to justify all answers. (a) Suppose A, B, C and D are n x n square matrices, and the inverse of A exists. __ —1 J: On ’ F: A B , G: In A B ’ —CA-1 In 0 D On In i. (1 point) Find JFG. ii. (2 points) Show that det < 2’ g > = (det A) (det (D—CA’IB». (b) (3 points) Suppose A be an n X n invertible matrix. Let adj A be the adjoint of A. Show that (adj A)T = adj (AT) (0) (3 points) Let A be an m X 71. matrix Whose columns are linearly dependent. Suppose that b belongs to the column space of A. Show that Ax = b has infinitely many solutions. (d) (3 points) Suppose that the n x 71, matrix A satisfies A2 = A. Show that the only possible eigenvalues of A is O and 1. (e) (3 points) Let S be the subspace of R4 spanned by 1 2 —-1 —2 2 5 ——2 —3 T Find the distance from x = [ 2 0 3 —5 ] to 5*. ...
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MATH 2510A Final 0910 - 1¥~§ (£- 3 E) Pagelof 3 : m v 7s...

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