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MAT331-2003Spring

MAT331-2003Spring - Final Examination MAT 331 Spring 2003...

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Unformatted text preview: Final Examination MAT 331, Spring 2003 Do all the problems. Show work and proper reasons to get credit. 1. (11 points) Determine whether the system of linear equations are consistent. If it is consistent, ﬁnd the solutions(s). If there are multiple solutions, write them in parametric vector form (no other form). (3) I1 "l" 11:2 + 1'3 “ IE4 : 2 1‘2 "l’ 2333 — 2.734 2 3 {131 + SE3 “ 134 : 1 \$1 + \$2 : 2 (b) 171 + 2332 — 333 — 22:4 : 4 (131 + 2.732 + \$3 2 6 2 (8 points) Find a basis of the subspace spanned by the vectors 1 1 1 1 2 '01: 1 7112: 2 ,U3: 3 ,U4= 5 ,'U5: 3 2 3 4 7 2 3 (9 points) (a) Find the inverse of the matrix 1 O 2 1 0 1 v—Imi—I (b) Use the inverse in part (a) to ﬁnd the solution of 1 0 1 2:1 1 2 l 2 1132 = 2 4 (8 points) Find the a basis of the null space of the matrix 1 —2 —3 1 —2 —3 1 —2 —3 5 (6 points) Find the determinant of the matrix l—‘OOOp—I OOONO OOOOOO OJAOOO UIOOOv—l 6 (9 points) (a) Draw the image of the triangle S, with vertices (0,0), (1,0), and (1, 1), under the linear transformation 3 1 T = l 1 2 l (b) Find the area of the image T(S). 7 (8 points) Find the area of the parallelogram whose vertices are (1, 1), (4,2), (2,4), and (5,5). 8 (9 points) Let p1(t) = 1 +t2, 102(15): 2 —t—t2, p3(t):1+ 2t — 4t? (a) Show that B = {p1,p2,p3} form a basis of 11%. (b) Find the polynomial p(t) in 1% whose coordinates relative to the basis [3 is given by 1 [P13 2 9 (8 points) Let 1 3 1 y: 1 ,u: ~1 ,1): —1 1 2 —2 Find the orthogonal projection of y onto the subspace W = span{u,v}. 10 (8 points) Find all the eigenvalues of the matrix 3 3 1 0 0 —1 WNW 11 (8 points) /\ = 3 is an eigenvalue of the matrix 4 2 3 —1 1 —3 2 4 9 Find a basis of the eigenspace corresponding to /\ = 3. 12 (8 points) Let —2 2 A - l —s s l Find a matrix P and a diagonal matrix D such that A = PDP’l. ...
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MAT331-2003Spring - Final Examination MAT 331 Spring 2003...

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