WA 4
Submit this homework assignment on Wed, Feb. 26, 2014.
Problem 1. Use Rolles theorem to discuss the position and multiplicity of
r
2
n
zeros for d (x 1) for r = 0, 1, . . . and hence show that the zeros of the Legendre
dxr
polynomial Pn discussed in
Final review problems: Solutions
Problem 1. Given matrix A and its row reduced form R: 1 2 0 1 0 -10/7 5 , R = 0 1 5/7 . A= 0 7 3 -1 -5 0 0 0 (a) Find a basis and the dimension for the column space of A; (b) Find a basis and the dimension for the row spac
Exam 2: Solutions
Problem 1. (a) Calculate the determinant by a cofactor expansion.
At each step, choose a row or column that involves the least amount of
computation:
14
0
7
5
0
0 7 3 5
02
0
0
3 6 4 8 ;
05
2 3
0 9 1 2
(b) Calculate the determinant using
HW2: Solutions to some problems
E 2.2.4. We rst observe that
n
2
1
(Tn f )(x) =
2
k=n
0
1
f (t)ek (t) dt ek (x) =
2
n
2
eik(xt) dt
f (t)
0
k=n
=
1
2
2
f (t)Dn (x t) dt,
0
i.e., Tn f = Dn f , and
n
n
Dn (x) =
eikx =
ek (x) =
k=n
k=n
ei(n+1/2)x ei(n+1/2)x
e
WA3: Solutions to some problems
E 2.3.2. Dene the mapping T :
1
(c0 ) by
x, T y =
xn y n .
n=1
It is easy to check that T is linear and
x
1
x
1
|xn | |yn | sup sup |xm |
T y = sup | x, T y | sup
x
n=1
1
m
|yn | = y 1 .
n=1
(n)
xk
Thus T B ( 1 , (c0 )
Final Exam - MATH 640
December 6, 2013
Name
Solve any 5 out of 6 problems and get up to 35 grade points (i.e.,
each of the selected problems is worth 7 grade points). An additional
problem is worth 5 extra credit points. Please, mark the extra credit
prob
Exam 1: Solutions
Problem 1. Given that
11
b = 5 ,
9
1 2 6
7 ,
A = 0 3
1 2 5
(a) determine whether b belongs to the span of the columns of A;
(b) if the answer in part (a) is yes, represent b as a linear combination
of the columns of A.
Solution. (a) The
Final review problems
Problem 1. Given matrix A and its row reduced form R: 1 2 0 1 0 10/7 5 , R = 0 1 5/7 . A= 0 7 3 1 5 0 0 0 (a) Find a basis and the dimension for the column space of A; (b) Find a basis and the dimension for the row space of A; (c) Fi
Final Exam - Solutions
2
2
Problem 1. (a) Evaluate B ex y dxdy , where B consists of those
points (x, y ) R2 which satisfy x2 + y 2 1 and y 0.
(b) Evaluate
S
(x2
dx dy dz
,
+ y 2 + z 2 )3/2
where S is the solid bounded by the spheres x2 + y 2 + z 2 = a2 a
Final Exam - Solutions
2
2
Problem 1. (a) Evaluate B ex y dxdy , where B consists of those
points (x, y ) R2 which satisfy x2 + y 2 1 and y 0.
(b) Evaluate
S
(x2
dx dy dz
,
+ y 2 + z 2 )3/2
where S is the solid bounded by the spheres x2 + y 2 + z 2 = a2 a
WA 3
Submit this homework assignment on Monday, February 17.
Problem 1. Let f be a continuous function on T and / an irrational number.
Prove that
N
1
1
f (t) dt
f ( + n) =
lim
N N
2
n=1
for every T. (Hint: Prove it rst for f () = eik , k = 0, 1, 2, . .
WA 5
March 1, 2014
Submit this homework assignment on Friday, March 7, 2014.
Problem 1. Let f (x) = ex for x 0, f (x) = 0 for x < 0. Write
f1 = f and fn = fn1 f (n 2).
(a) Find fn explicitly for all n 1.
(b) Verify by direct computation that fn () = f ()n
Practice Exam 2
Problem 1. (a) Calculate the determinant using cofactor expansions.
At each step, choose a row or column that involves the least amount of
computation:
1 3
0
0
2 6
1
0
12
20
.
75
3 4
(b) Calculate the determinant using row reduction:
1
3
3
Practice Exam 1
Problem 1. Let
1
3
0
3
1 1 1 1
A=
0 4 2 8 .
2
0
3 1
(a) Describe all vectors
b1
b2
b=
b3
b4
that belong to the span of the columns of A in terms of equations which
involve b1 , b2 , b3 , b4 .
(b) Does the vector
1
2
b=
3
4
belong to
WA 1
January 24, 2014
Submit this homework assignment on January 31.
Problem 1. Let the sequence of complex numbers cfw_sn be dened as sn =
(1)n (2n + 1), n = 0, 1, . . . Show that the sequence of its Cesaro means n =
n
1
j =0 sj does not have a limit, h
WA 2
Submit this homework assignment on Friday, February 7.
Problem 1. By considering the Fourier sum associated with the function f :
T C given by f (t) = eit for a non integer C and 0 < t < 2 (the value of
f at 0 = 2 doesnt matter for the calculation of
Final Exam: Solutions
Problem 1. (a) Find all continuous functions f on the line segment
[0, 1] which satisfy
1
f (x)xn dx = 0,
n = 0, 1, . . .
0
(b) What will be the answer if one requires only that
1
f (x)xn dx = 0,
n = 2014, 2015, . . .?
0
1
Solution.
Exam 1 - MATH 321: Solutions
Problem 1. (a) Can the function f (x, y ) =
uous by suitably dening it at (0, 0)?
(b) The same question as in (a) for f (x, y ) =
xy
x2 + y 2
be made contin-
sin(x2 +y 2 )
.
x2 + y 2
Solution. (a) The partial limits of f along