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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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This test prep was uploaded on 04/10/2008 for the course MAT 331 taught by Professor Li during the Fall '07 term at Syracuse.
 Fall '07
 Li
 Linear Algebra, Algebra

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