Name
Math 311
Final Exam Version A
Spring 2015
Section 502
P. Yasskin
1-8
/40
10
/20
9
/16
11
/28
Total
/104
Multiple Choice: 5 points each. No Part Credit
Work Out: Points indicated. Show all work.
1. Hydrocloric acid H Cl and sodium hydroxide NaOH react to produce sodium chloride Na Cl and
water H 2 O according to the chemical equation:
aH Cl bNa O H
cNa Cl
dH 2 O
Which of the following is the augmented matrix which is used to solve this chemical equation?
(Put the elements in the order H, Cl, Na, O.)
a.
c.
1
1
0
0
0
1
0
1
1
0
0
1
1
0
0
2
0
0
1
0
1 1 0 0
0
1 0 1 1
0
0 1 1 0
0
2 0 0 1
0
b.
d.
2. Suppose A is nilpotent, i.e. A 2
a.
b.
c.
d.
e.
A
A
A
A
A
1
1
1
1
1
is invertible and
is invertible and
is invertible and
is invertible and
is not invertible
A
A
A
A
1 1 0 2
0
1 0 1 0
0
0 1 1 0
0
0 1 0 1
0
1 1
0
2
0
1 0
1
0
0
0 1
1
0
0
0 1
0
1
0
0. Which of the following is true?
1
1
1
1
1
1
1
1
A 1
1 2A
1 A
1 A
1
3. Consider the vector space M 2, 2
1 0
E1
E2
0 1
of 2
1
0
0
1
2 matrices with the basis
0 1
E3
1 0
9 5
Which of the following are the components of the matrix A
a.
A
E
2 1 4 3
b.
A
E
5 4 3 2
c.
A
E
4 5 2 3
d.
A
E
1 2 3 4
e.
A
E
5
4 3
subspace V Span x, x 2 spanned by the basis v 1
orthonormal basis for V?
b. u 1
5 x
2
x
u2
c. u 1
d. u 1
e. u 1
u2
3 x
2
2x
1 0
relative to the E basis?
of real valued function on the interval
derivatives exist and are continuous with the inner product
x
1 1
1
2
4. Consider the vector space C 2 0, 1
a. u 1
0
E4
1
0, 1
whose second
f x g x dx. Consider the
f, g
1
x, v 2
x 2 . Which of the following is an
x2
7 x2
2
3
2
x
x
4
5 x2
u2
2
u2 4 5 x2 3 5 x
u2
2
5. Consider the second derivative linear operator L : P 6
P6 : L p
polynomials of degree less than 6. Find the image, Im L .
HINT: Let p a bx cx 2 dx 3 ex 4 fx 5 .
a.
b.
c.
d.
e.
Im
Im
Im
Im
Im
L
L
L
L
L
Span
Span
Span
Span
Span
P6 : L p
polynomials of degree less than 6. Find the kernel, Ker L .
HINT: Let p a bx cx 2 dx 3 ex 4 fx 5 .
7.
Ker
Ker
Ker
Ker
Ker
L
L
L
L
L
on the space of
d2p
dx 2
on the space of
1
1, x
1, x, x 2 , x 3
x2, x3, x4, x5
x, x 2 , x 3 , x 4 , x 5
6. Consider the second derivative linear operator L : P 6
a.
b.
c.
d.
e.
d2p
dx 2
Span
Span
Span
Span
Span
1
1, x
1, x, x 2
x2, x3, x4, x5
x, x 2 , x 3 , x 4 , x 5
Compute the line integral
F ds clockwise around the
y
3
complete boundary of the plus sign, shown at the right,
for the vector field
a.
25
b.
5
c. 0
d. 5
F
4x 3
2y, 4y 3
3x .
2
1
0
0
1
2
3
x
e. 25
3
8. Consider the vector space M 2, 2
of 2
2 matrices. Let A
function, L : M 2, 2
M 2, 2 : L X
AX
corresponding eigenmatrix (eigenvector) of L?
x y
HINT: Let X
z w
a.
2
X
b.
1
X
c.
0
X
d.
0
X
e.
1
X
1 0
. Consider the linear
0 2
XA. Which of the following is not an eigenvalue and
.
0 1
1 0
0 0
1 0
1 0
0 0
0 0
0 1
0 1
0 0
4
9.
16 points
Which of the following is an inner product on
Put ’s in the correct boxes. No part credit.
Let x
x1, x2 , y
y1, y2 .
2
? If not, why not?
Why not?
Inner Product?
Yes
x, y
a. x 1 y 1
2x 2 y 2
b. x 1 y 1
2x 1 y 2
c. x 21 y 21
2x 22 y 22
d. x 1 y 1
x1y2
x2y1
2x 2 y 2
e. x 1 y 1
x1y2
x2y1
2x 2 y 2
f.
x1y1
x1y2
x2y1
x2y2
g. x 1 y 1
x1y2
x2y1
x2y2
h. x 1 y 1
x2y2
No
Not
Not
Not
Positive but Not
Symmetric Linear Positive Positive Definite
2x 2 y 2
5
10.
20 points
Let M 2, 3 be the vector space of 2 3 matrices.
Consider the subspace V Span A 1 , A 2 , A 3 , A 4 where
A1
1 1
0
2 3
2
, A2
2 2
0
4 6
4
, A3
1 0 2
3 0 0
, A4
1 2
6
5 6
4
Find a basis for V. What is the dim V ?
6
11.
28 points
Compute
F dS over the hemisphere z
25
x2
y2
oriented upward, for the
H
vector field F
x 3 4y 2 4z 2 , 4x 2 y 3 4z 2 , 4x 2 4y 2
HINT: Use Gauss’ Theorem by following these steps:
z3 .
a. Write out Gauss’ Theorem for the Volume, V, which is the solid hemisphere
0 z
25 x 2 y 2 . Split up the boundary, V, into two pieces, the hemisphere, H, and
the disk, D, at the bottom. State orientations. Solve for the integral you want.
b. Compute the volume integral using spherical coordinates.
(continued)
7
c. Compute the other surface integral over D by parametrizing the disk, computing the tangent
vectors and normal vector, checking the orientation, evaluating the vector field on the disk and
doing the integral.
d. Solve for the original integral.
8