Homework Assignment 11
Assigned Fri 04/22. Due Fri 04/29.
1. Let V be an inner product space.
(a) If U V is a subspace, show that U (U ) .
(b) If dim(V ) < , show that U = (U ) .
2. Suppose we want to nd a line y = c0 + c1 x that best ts the points (0, 0)
21341 Linear Algebra: Final.
Thu 05/05, 2011
This is a closed book test. No calculators or computational aids are allowed.
You have 3 hours. The exam has a total of 9 questions and 90 points.
You may use without proof any result that has been proved in
Math 341 Final.
Tue, Dec 15, 2009
Time: 50 mins
Total: 80 points
This is a closed book test, and your are not allowed to use calculators or other computational aids. You may use any result from class/homework provided you make an appropriate
reference, un
Math 341 - Linear algebra
FINAL EXAM
Monday Dec. 13, 2010
1. (2/100) Name:
2. (10/100) Suppose that V is a vector space with complex inner product , . Prove that for all v, w V
v, w =
where i is the imaginary constant (i2 = 1).
1
4
4
ik v + ik w 2 ,
k=1
3
Homework Assignment 10
Assigned Fri 04/08. Due Fri 04/22.
This homework is a double homework in order to avoid a conict with Spring carnival. It will
count as two homeworks for your grade. The second half of your homework will be about inner
product space
21341 Linear Algebra: Midterm 1.
Wed 02/09
This is a closed book test. No calculators or computational aids are allowed.
You have 50 minutes. The exam has a total of 5 questions and 50 points.
You may use without proof any result that has been proved i
Homework Assignment 9
Assigned Fri 04/01. Due Fri 04/08.
1. (Optional) Let M be an upper triangular matrix. Show that the eigenvalues of M are exactly the
diagonal entries of M . [I stated, but did not nish proving this in class.]
1
1
.
.
2. (a) Let M =
Math 341 - Linear algebra
Midterm 2 (A)
Wed. Oct. 27, 2010
1. (2/50) Name:
2. (10/50) Suppose that V is vector space with (real) inner product , . Prove that for all u, v V
u, v =
1
[ u+v
4
2
u v 2 ].
3. Suppose that T L(R2 , R3 ) is such that
T
1
1
1
Solutions to Homework Assignment 12
Assigned Fri 04/29. Due never.
2. Let V be a nite dimensional inner product space. Show that T L(V, V ) is
positive if and only if there exists S L(V, V ) such that T = S S.
2
1. Let a, b, c R, and dene C R by
C = cfw_(
21341 Linear Algebra: Midterm 1.
Wed 03/23
This is a closed book test. No calculators or computational aids are allowed.
You have 50 minutes. The exam has a total of 5 questions and 50 points.
You may use without proof any result that has been proved i
Math 341 Syllabus and Lecture schedule.
L10, Wed 02/02.
Gautam Iyer, Fall 2010
L1, Mon 01/10.
Error correcting codes.
F = cfw_0, 1, choose a, b, c F .
Pick cfw_v1 , v2 , v3 F n , and transmit the message u =
av1 + bv2 + cv3 .
Let C = spancfw_v1 , v2
LINEAR ALGEBRA HOMEWORK 4 SOLUTIONS
JAMES CUMMINGS
This homework is due by class time on Friday 21 February. You may not collaborate. Ask me if you nd that a question is unclear or suspect that you have
found a typo. Your solutions must either be word-pro
LINEAR ALGEBRA HOMEWORK 4
JAMES CUMMINGS
This homework is due by class time on Friday 21 February. You may not collaborate. Ask me if you nd that a question is unclear or suspect that you have
found a typo. Your solutions must either be word-processed or
LINEAR ALGEBRA HOMEWORK 3 SOLUTIONS
JAMES CUMMINGS
(1) Let V be a vector space over a eld k and let : V V be a linear
transformation. We dene powers of by the recursive denition 1 =
, t+1 = t (if you prefer something less formal, t is obtained by
composin
LINEAR ALGEBRA HOMEWORK 1 SOLUTIONS
JAMES CUMMINGS
This homework is due by class time on Monday 27 January. You may not
collaborate. Ask me if you nd that a question is unclear or suspect that you have
found a typo. Your solutions must either be word-proc
LINEAR ALGEBRA HOMEWORK 1
JAMES CUMMINGS
This homework is due by class time on Monday 27 January. You may not
collaborate. Ask me if you nd that a question is unclear or suspect that you have
found a typo. Your solutions must either be word-processed or w
LINEAR ALGEBRA HOMEWORK 2 SOLUTIONS
JAMES CUMMINGS
Observation: A linear map is injective if and only if its nullspace is zero,
because (v1 ) = (v2 ) if and only if v1 v2 is in the nullspace.
(1) Let V and W be vector spaces over a eld k and let : V W be
LINEAR ALGEBRA HOMEWORK 3
JAMES CUMMINGS
This homework is due by class time on Friday 14 February. You may not collaborate. Ask me if you nd that a question is unclear or suspect that you have
found a typo. Your solutions must either be word-processed or
LINEAR ALGEBRA HOMEWORK 2
JAMES CUMMINGS
This homework is due by class time on Friday 7 February. You may not collaborate. Ask me if you nd that a question is unclear or suspect that you have found
a typo. Your solutions must either be word-processed or w
LINEAR ALGEBRA HOMEWORK 5
JAMES CUMMINGS
You may consult any printed or online source, but must cite all sources other
than the textbook. Contact me if you think you have found a typo or do not
understand a question. All questions carry equal weight. This
Math 341 - Linear algebra
Midterm 1 (A)
Wed. Sept. 22, 2010
1. (2/50) Name:
2. (10/50) Suppose that R is such that 2 Q. Let
F := cfw_a + b : a, b Q.
1
Prove that if x F and x = 0 then x F . Note: The rest of properties of a eld are easy to verify,
and hen
Homework Assignment 1
Assigned Mon 01/10. Due Fri 01/14.
1. Suppose F (with binary operations +, ) is some eld, and assume all quantities referenced in this
problem are elements of F . Do this problem using only the basic eld axioms.
(a) Show that the add
10
Math 341 Homework (Fall 2009)
Assignment 14: Assigned Sat 12/05. Due Never.
1. (a) Show that (T ) = T .
(b) Show that (ST ) = T S .
2. Let S, T L(V, V ) be such that ST = T S.
(a) If S and T are diagonalisable, show that there exists a basis of V consi
Math 341 Homework (Fall 2009)
9
Assignment 13: Assigned Wed 11/25. Due Fri 12/04. Last ever
1. The following statements were stated, but not completely proved in class. Prove them.
(a) If v1 , . . . , vn are eigenvectors of T corresponding to n distinct
8
Math 341 Homework (Fall 2009)
Assignment 12: Assigned Wed 11/18. Due Tue 11/24 by 10:00AM
Please note the due date is Tuesday at 10:00AM because there will be no class the Wednesday
before Thanksgiving. You may turn in your homework in class on Monday 1
Math 341 Homework (Fall 2009)
Assignment 1: Assigned Wed 08/26. Due Wed 09/02
1. Suppose F (with binary operations +, ) is some eld, and assume all quantities referenced in this
problem are elements of F . Do this problem using only the basic eld axioms.
LINEAR ALGEBRA FINAL: TAKE HOME PART
JAMES CUMMINGS
This part of the nal is due by 8:30am on Mon Dec 18. You may work on it
during any continuous 24 hour period of your choice. You may not collaborate.
You may consult your notes and any other material you
Math 341 Homework (Fall 2009)
7
Assignment 11: Assigned Wed 11/11. Due Wed 11/18
1. In class we showed that if A is an invertible n n matrix, then |AB| = |A|B| for any n n matrix
B. This problem proves |AB| = |A|B| if A is not invertible.
If A, B are two