Name
Math 311
ID
Final Exam
Spring 2002
Section 503
P. Yasskin
x
1.
10 points Extra Credit
Determine if and where the line
intersects the plane
y
z
2
=
y
=
−8
3
1
+r
−1
2
/10 3
/30
3
/40 4
/30
1
z
x
1 EC
2
3
4
+t
5
6
3
+s
6
.
4
1
2 1
2.
40 points
Let Mp, q  be the vector space of p × q matrices, and let P =
4 2
.
6 3
Consider the linear function
L : M2, 2 
a. 5  Let X =
a b
c d

M3, 2  given by LX  = PX
and compute LX .
b. 2  Identify the domain of L. What is its dimension?
c. 2  Identify the codomain of L. What is its dimension?
d. 8  Identify the kernel (null space) of L. Give a basis and the dimension.
2
Problem 2 continued:
e. 8  Identify the image (range) of L. Give a basis and the dimension.
f. 3  Is L onto? Why?
g. 3  Is L one-to-one? Why?
h. 2  Check that the dimensions of the kernel and image are consistent with the dimensions of the domain and
codomain.
3
Problem 2 continued:
i. 7  Find the matrix of L relative to the bases
E1 =
1 0
0 0
E2 =
0 1
0 0
1 0
F1 =
F4 =
0 0
E3 =
0 0
1 0
0 0
F2 =
1 0
E4 =
F3 =
0 0
0 0
1 0
0 1
0 0
0 0
0 0
F5 =
0 1
0 0
0 1
0 0
0 0
0 0
0 0
F6 =
0 0
0 1
4
3.
30 points On the vector space P 2 =
polynomials given by
polynomials of degree less than 2
⟨p, q ⟩
a. 15  Show
⟨p, q ⟩
consider the function of two
∞
= ∫ px qx e −x dx
0
is an inner product.
b. 15  Find the angle θ between the polynomials px  = 1 and qx  = x.
You may use these integrals without proof:
∞
∞
∫ e −x dx = 1, ∫ xe −x dx = 1,
0
0
∞
∫ x 2 e −x dx = 2,
0
∞
∫ x 3 e −x dx = 6
0
5
4.
30 points Gauss’ Theorem states that if V is a volume in R 3 and ∂V is its boundary surface oriented
⃗ is a nice vector field on V then
outward from V and F
⃗ dV = ∫∫ F
⃗ ⋅ dS
∫∫∫ ⃗∇ ⋅ F
V
∂V
⃗ = xz 2 , −yz 2 , x 2 z + y 2 z 
Verify Gauss’ Theorem if F
and V is the volume above the paraboloid
z = x 2 + y 2 and below the plane z = 9.
Notice that ∂V consists of the paraboloid P
and a disk D.
Be sure to use the correct
orientations for P and D.
⃗ dV :
a. 6  Compute ∫∫∫ ⃗
∇⋅F
V
⃗=
∇⋅F
i. ⃗
⃗ dV =
ii. ∫∫∫ ⃗
∇⋅F
V
6
Problem 4 continued:
⃗ ⋅ dS :
b. 10  For the paraboloid P compute ∫∫ F
⃗ = xz 2 , −yz 2 , x 2 z + y 2 z .
Recall F
P
⃗ r, θ  =
i. R
⃗r =
ii. R
⃗θ =
iii. R
⃗=
iv. N
⃗ R
⃗ r, θ 
v. F
⃗ R
⃗ r, θ 
vi. F
=
⋅
⃗=
N
⃗ ⋅ dS =
vii. ∫∫ F
P
7
Problem 4 continued:
⃗ ⋅ dS :
c. 10  For the disk D compute ∫∫ F
⃗ = xz 2 , −yz 2 , x 2 z + y 2 z .
Recall F
D
⃗ r, θ  =
i. R
⃗r =
ii. R
⃗θ =
iii. R
⃗=
iv. N
⃗ R
⃗ r, θ 
v. F
⃗ R
⃗ r, θ 
vi. F
=
⋅
⃗=
N
⃗ ⋅ dS =
vii. ∫∫ F
D
d. 4  Verify the two sides of Gauss’ Theorem are equal.
8