mat188f_2005_exam - UNIVERSITY OF TORONTO FACULTY OF...

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Unformatted text preview: UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, DECEMBER 2005 First Year — CHE, CIV, IND, LME, MEC, MMS MAT188H1F — LINEAR ALGEBRA Exam Type: A Examiners SURNAME D. Burbulla GIVEN NAME F. Latremoliere V. Litvinov STUDENT NO. S. Uppal SIGNATURE Calculators Permitted: Casio 260, Sharp 520 or Texas Instrument 30 INSTRUCTIONS: Attempt all questions. MARKER’S REPORT Page 2 Page 5 Page 8 Questions 1 through 6 are Multiple Choice; circle the single correct choice for each question. Each correct choice is worth 4 marks. Question 7 consists of twelve statements which you must show are True or False; 2 marks each. Questions 8 through 11 are 10ng questions for which you must show your work. Each 10ng question is worth 13 marks. TOTAL MARKS: 100. Use the backs of the pages if you need more space. Page 9 Page 1 of 9 1. If U is a subspace of R5 and dim U = 2, then dim U ‘L is (a) 2 (I: y ] to a solution of the inconsistent system of 3. The best approximation Z = [ equations 14 —14 0 rah l\') 8 8 + 03 fifi Illl Page 2 of 9 4. What is the matrix of the transformation which is composed of a reflection in the :c-axis followed by a rotation through g? 1 5. The eigenspaces of the projection matrix Pm +mzlm1 m m2] are (a) E_1(Pm) = spanum 1 l} and 571(1):;“llmll (b)Eo<P m>=spanllm } lm } Mmpnmfllnmmpswmfl;]} [ haw—spat l} and E1(Pm) = span { ] ——m <d>E1(Pm> >=span{ 1 6. The equation of the plane passing through the point (cc,y, z) = (1,0, —1) and perpendicular to the line [ac y z]T = [2 3 4]T + t[2 1 3]T (a) 2x+3y+4z=——2 (b) 2m+y+3z=1 (c) 2$+3y+4z=2 (d) 2x+y+3z=~1 Page 3 of 9 7. Suppose A is an n X n matrix such that A2 = 0. Explain clearly and concisely Why the following six statements about A are True. (a) det(A) = 0 (b) (ATV = 0 (c) (I—A)"1=I+A (d) If the system AX = B is consistent, then B is in nullA, where X and B are n x 1 matrices. (e) The only eigenvalue of A is A = 0. (f) If A is diagonalizable then A = O Page 4 of 9 7. (continued) Suppose A is an n x n matrix such that A2 = 0. Explain clearly and concisely Why the following six statements about A are False. (QA=0 (h) adj(A) = 0 (i) A is invertible (j) colA = nullA (k) imA = R” (1) dim(Eo(A)) = n Page 5 of 9 8. Given that 1 O 1 —1 1 1 0 0 0 10 2 0 3 1 1 0 0 1 0 —7 A — 1 O 0 _4 2 has reduced row—echelon form R —- 0 0 0 1 2 , 0 0 1 4 1 ‘ 0 0 0 0 0 state the rank of A, and then find a basis for each of the following: the row space of A, the column space of A, and the null space of A. Page 6 of 9 9. LetAbe annxnmatrix;letU={XinR"IAX=ATX}. (a) [5 marks] Show that U is a subspace of R". 12 3 4 1 6 3 4 5 3 7 (b)[8 marks] Let A = 3 5 6 4 8 . Find a basis for U. 1 3 5 2 9 6 7 8 9 1 Page 7 of 9 10. LetU=span{[0 1 0 0]T,[1 —1 —1 1]T,[1 2 —2 0F}; let X = [ 1 1 0 1 ]T. Find projU(X) and projU;(X). Page 8 of 9 11. Find an orthogonal matrix P and a diagonal matrix D such that D if 0 O 1 A: 0 —1 O . 1 0 0 Page 9 of 9 PTAP, ...
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