Name
ID
MATH 311
Final Exam
Section 200
1.
Spring 2001
1
/30 4
/20
2
/10 5
/10
3
/15 6
/15
P. Yasskin
30 points Let SŸ2, 2 be the set of 2 • 2 symmetric matrices, i.e. 2 • 2 matrices M satisfying M T M.
Consider the function L : MŸ2, 2 ¯ SŸ2, 2 given by LŸX X X T .
a. Ÿ5 Show that SŸ2, 2 is a subspace of MŸ2, 2 , the vector space of 2 • 2 matrices.
5 Find a basis for SŸ2, 2 . What is the dimension of SŸ2, 2 ?
b.
Ÿ c.
Ÿ 5 Show L is linear.
1
d.
Ÿ
15 For the linear function L, identify
S
1 DomŸL Ÿ dim DomŸL S
1 CoDomŸL Ÿ dim CoDomŸL S
3 KerŸL Ÿ dim KerŸL S
3 RanŸL Ÿ dim RanŸL S
3 1 " 1?
Ÿ Circle:
Yes
No
Circle:
Yes
No
Why?
S
3 onto?
Ÿ Why?
S
1 Verify the Nullity-Rank Theorem for L.
Ÿ 2
2.
10 points
Consider the function of two matrices X a b
c d
and Y p q
r s
given by
X, Y ™ trŸXY ˜
where tr means ”trace” which is the sum of the principle diagonal entries, i.e. tr
Explain why
,
w x
y z
is an inner product on SŸ2, 2 , but is not an inner product on MŸ2, 2 .
w z.
3
pŸ"1 3.
15 points
Consider the linear map L : P 2
¯
R given by LŸp 3
pŸ0 .
pŸ 1 a.
5 Find the matrix of L relative to the bases e £e 1 1, e 2 t, e 3 t 2 ¤ for P 2 and
1
0
0
,i 2 ,i 3 i i 1 for R 3 . Call it A.
0
1
0
Ÿ 0
b.
1
5 Find the matrix of L relative to the bases q £q 1 1 t 2 , q 2 t t 2 , q 3 t 2 ¤ for P 2 and
0
1
2
v 1 , v 2 , v 3 v
for R 3 . Call it B.
0
0
1
Ÿ 1
c.
0
1
2
5 Find the matrix B by a second method.
(© Use opposite page.)
Ÿ 4
4.
20 points Consider the helix H parametrized by rŸt Ÿ4 cos t, 4 sin t, 3t between A Ÿ4, 0, 0 and
B Ÿ"4, 0, 3= .
a.
Ÿ
B
ds of the vector field F
Ÿyz, "xz, z along the helix H.
10 Compute the line integral ; F
A
H
b.
Ÿ
10 Find the total mass of the helix H if the linear mass density is > z 2 .
5
10 points
5.
Compute > x dx z dy " y dz around the boundary of the triangle with vertices Ÿ0, 0, 0 ,
0, 1, 0 and Ÿ0, 0, 1 , traversed in this order of the vertices.
Ÿu, v Ÿ0, u, v .
HINT: The yz-plane may be parametrized as R
Ÿ
6
6.
15 points
Gauss’ Theorem states
F
dS
dV ;; F
;;; V
V
where V is the total boundary of V with OUTWARD normal.
Let V be the solid cone x 2 y 2 t z t 2.
Let C be the conical surface z 2
x 2 y 2 for z t 2
with UPWARD normal.
-2
Let D be the disk x 2 y 2 t 4 with z 2
0
r
2
-2
2
with UPWARD normal.
dS for F
Ÿxy 2 , yx 2 , z 3 in two ways.
Compute ;; F
C
a.
dS explicitly.
5 Method I: Parametrize C and compute ;; F
Ÿ C
7
b.
Ÿ
F
dS and ;;; dS.
dV and solve for ;; F
10 Method II: Parametrize D and V, compute ;; F
D
V
C
Be very careful with the orientation of the surfaces.
8