MATH 2510A Final 0708

MATH 2510A Final 0708 - g”? (iii-3 E) Pagelof 3 : misma-...

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Unformatted text preview: g”? (iii-3 E) Pagelof 3 : misma- Tamra] jak /% Ci: i j: 3% Copyright Refsemed The Chinese University of Hong Kong ;0 0i 5 ;0 0A 5H)? .J': $3flfii37¥€it Course Examination fl Term, 200i - 200 _8_ it a 31% tit/3U; 7% course code & Title ; MAT2310A LINEAR ALGEBRA AND APPLICATIONS 33} Fall ’1‘ 33} 9} ~ E Time allowed : ______________ _______________ __ hours ____________ __________ __ minutes Student ID. No. : Seat N0. : Answer all questions. 1. Let —1 O —1 1 0 1 —1 2 A: 2 2 O 2 2 ——1 3 1 0 —1 1 0 (a) (5 marks) Find a reduced row echelon form U of A. (b) (5 marks) Find a basis for the row space of U and a basis for the row space of A. Explain your answer. (c) (5 marks) Exhibit some columns of A that form a basis for the column space. What is the dimension of the column space? (d) (5 marks) Find a basis of the null space of A. What is the nullity of A? 2. Let T : R4 —+ R3 be the linear transformation defined by —2$1 + 2552 + 6333 + 4T4 T = 4331 ~ 2583 — 2x4 3331 + 4x; + 5mg, + 8x4 (a) (7.5 marks) Determine whether T is one-to—one. Justify your answer. (b) (7.5 marks) Determine whether T is onto. Justify your answer. Course Code f} E 3 a ;_ E (at; 3 E) Page 2 of 3 3. Let i—Ar—AOr—A [\3 (a) (8 marks) Use the Gram~Schmidt process to find an orthonormal basis for the column space of A. (b) (5 marks) Find a QR factorization of A. (c) (2 marks) Use 3(b) to find the least squares solution IE1 — 1'2 2 1 + 2552 2 $1 + $2 ‘— _1 $1 — 31132 — 3 Find the matrix A representing L with respect to the bases 1 1 U=w2}={l—2H1l} V = {V1,V2,V3} = and M Course Code 2?!- E é‘fiai’fi 3 ............................. .............................. .. fl 3 E (éé 3 E) Page 3 of 3 5. Let 5 4 —4 A = 4 5 4 —4 4 5 (a) (17 marks) Show that the matrix A is diagonalizable. Hence, find P such that P‘lAP is a diagonal matrix. Justify your answers. (b) (3 marks) Then calculate the matrix K = %A3 — A—1 — 32]. 6. Recall that P3 is the vector space of all polynomials p(t) of degree at most 3. Let us consider the subset S of P3 consisting of polynomials p such that 19(2) 2 0. In other words, the polynomial p is in S if 2 is a root of p. (a) (5 marks) Show that S is a vector subspace of P3. (b) (5 marks) Determine a basis for S. Justify that your answer is a basis. What is the dimension of S? 7. Are the following statements true or false? You will not receive any mark unless you justify your answers. (a) (2.5 marks) The equation Ax = b is consistent if the augmented matrix [Alb] has a pivot position in every row. (b) (2.5 marks) If H and K are subspaces of a vector space V, then their union H U K is a subspace of V. (c) (2.5 marks) If B is a symmetric matrix and if Bv = 2V and Bu 2 —u then v - u = O. 7 (d) (2.5 marks) If the square matrix A is invertible, then it is diagonalizable. N End of Examination ~ ...
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MATH 2510A Final 0708 - g”? (iii-3 E) Pagelof 3 : misma-...

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