1. A
2.
a.
A= cfw_O , L , P
B=cfw_ O , L , P
C=cfw_ O , M
b.
A B
B A
c.
A C=cfw_ O
This is the set of cities with at least 2,000,000 jobs in 2000 or projected in 2025 and with
projected annual growth rate of a least 2.6%.
d. This is the set of cities
Math 562 - Homework 3
Solutions
Prof. Arturo Magidin
1. Let V = C3 with the usual inner product. Verify that [b1 , b2 , b3 ] are an orthogonal (not
necessarily orthonormal) basis, where b1 = (1, 1, 1), b2 = (2, 1, 1), and b3 = (0, i, i). Then use
inner pr
Math 562 - Homework 2
Solutions
Prof. Arturo Magidin
1.
(i) Apply the Gram-Schmidt orthonormalization process to S in the inner product space R3
(equipped with the standard inner product) to obtain an orthonormal basis :
S = (1, 0, 1), (0, 1, 1), (1, 3, 3
Math 562 - Homework 1
Solutions
Prof. Arturo Magidin
1. Let V be an inner product space over F . Prove the polar identities :
(a) If F = R, then x, y = 1 x + y 2 1 x y 2 .
4
4
Proof. Use the fact that v 2 = v, v and expand the right hand side:
x+y
2
xy
2
MATH 562
FINAL EXAM SOLUTIONS
Prof. Arturo Magidin
1. Find the singular values of the matrix A, where
1
0
A=
1
1
(10 points)
1
1
.
0
1
Solution. The singular values of A are the square roots of the nonzero eigenvalues of A A. So
rst we compute A A:
11
32
MATH 562 Linear Algebra and Applications II
Minimizing dissonance in a plucked string
We saw in class that the solution to the wave equation
2f
1 2f
2 2 = 0,
x2 v t
on [0, L], with Dirichlet boundary conditions f (0, t) = f (L, t) = 0 for all t; and init
MATH 562 MIDTERM
Solutions
Prof. Arturo Magidin
1. Let V be a nite dimensional complex inner product space, and let T be a linear operator on V.
For each of the following types of operators, give a property that characterizes that type in terms
of the exi
Math 562 - Homework 9
Solutions
Prof. Arturo Magidin
1. We found in the last homework the following singular value decomposition:
2
i 22
1+i
1
6 0 3(1i)
2
3
=
A=
1 i i
00
2
3
i 2
2
2
3
3
3
3(1+i)
3
Use this decomposition to nd the pseudoinverse A of A.
.
Math 562 - Homework 8
Solutions
Prof. Arturo Magidin
2
1. Let V = W = span(1, sin x, cos x) with inner product f , g = 0 f (t)g (t) dt. Let T : V W
be given by T (f ) = f + 2f . Find orthonormal bases = [v1 , . . . , vn ] and = [w1 , . . . , wm ]
for V an
Math 562 - Homework 7
Solutions
Prof. Arturo Magidin
1. Find a solution to the wave equation (with L = v = 1) with initial condition
f (x, 0) = 0
f (x, 0) =
t
and
1
if x 2 ;
1
if 2 < x 1.
x
1x
(This corresponds to a string which is struck, such as a piano
Math 562 - Homework 6
Solutions
Prof. Arturo Magidin
1. Let T be a linear operator on a complex inner product space, not necessarily nite dimensional,
and assume that T has an adjoint T . Prove:
(a) If T is self-adjoint, then T (v), v is real for all v V.
Math 562 - Homework 5
Solutions
Prof. Arturo Magidin
1. Find the plane z = a + bx + cy that has the best t (in the sense of least squares) to the points
(0, 0, 0),
(0, 1, 0),
Answer. Plugging in the values we obtain
b, and c:
(0, 0, 0) :
(0, 1, 0) :
(1, 0
1.
x< 2
2.
x<4
3.
( 4, )
4.
a.
8 x+ x 8
b.
8 xx
c.
8 x9
d.
8x
7
5.
y=7 x+ 14
6.
y=2 x4
7.
8
8.
a. 7.4051, -0.4051
b. 7.4051, -0.4051
a.
b. When average annual earnings for males is $45,000, average annual earnings for females
is $33,784.
c. 51334
9. NO S
1. 3, -6
2. 4, 1
3.
4.
a.
b.
c.
d.
(-1, -7)
Minimum
-1
-7
a. 80 units
b. 1180 $
5. 10, 90
6.
a.
b. (150, 3750)
c. Maximum
d. Positive
e. Negative
f. Closer to 0
7.
8.
a. Cubic
b. Quartic
a.
b.
c.
d.
9.
216
1
125
0.027
y=4 x5
10. Y= 8.2x 6.289
1.
a. Manufacturing constraint
Packaging constraint
b.
2.
3. Question is not clear
Radio= 60 mins
Television= 20 mins
4.
6 x+10 y 3000
x+ y 400
(x, y) = (3, 8)
F= 41
1.
a)
b) = 47.338 0.0142
c) 398.34 million people
d) () = 38.6 0.0263
2. DSG
a) = 0.14 0.6405
b)
The graph doesnt fit the data well.
c) 4 bats species
3.
a)
x
iny
2
4
-2.52573 -2.12026
6
8
10
12
14
16
-1.7148 -1.38629 -1.02165 -0.65393 -0.31471 0.058269
b
Math 562 - Homework 4
Solutions
Prof. Arturo Magidin
1. Let V = R3 with its usual inner product, and let L : V V be given by
L(a, b, c) = (3a + b, b c, 5a + c).
Find an expression for L (a, b, c).
Answer. The coordinate matrix of L relative to
3
[ L] = 0