Homework #2. Due Thursday, September 8th, in class
Important addition to homework policy: Problems marked with the
QD symbol (as opposed to Q symbol) are quiz problems which you are
allowed to discuss with others. However, the discussion should be limited
Solutions to selected homework problems.
Problem 2.1: Let p be any prime and V = Z2 , the standard twop
dimensional vector space over Zp . How many ordered bases does V have?
Answer: (p2 1)(p2 p).
Solution: First, by Corollary 3.5(c) any basis of V has tw
Solutions to selected problems in homeworks 9-11 (to be continued).
Problem 9.7: Find the Jordan canonical
matrix
210
0 2 1
A=
0 0 3
010
form and a Jordan basis for the
0
0
.
0
3
Solution: Step 1: we compute eigenvalues and multiplicities. Using the
formu
Algorithm for computing a Jordan basis (from Lecture 27).
Denition: Let V be a vector space, U a subspace of V and a subset of
V . We will say that
(a) is linearly independent mod U if is linearly independent and Span( )
U = cfw_0.
(b) is a basis for V /U
Math 5651. Fall 2011. Solutions to the in-class part of the second midterm.
1. Let V be a nite-dimensional vector space and n = dim(V ). Let S, T
L(V ) be s.t.
ST = 0
Prove that
rk(S ) + rk(T ) n
referring only to results proved in class.
Solution: Since
NAME:
Math 5651. Advanced Linear Algebra. Fall 2011. Second midterm (in-class part)
Tuesday, November 8th, 2-3:20 pm
Directions: No books, notes, calculators, laptops, PDAs, cellphones, web
appliances, or similar aids are allowed. All work must be your in
Advanced Linear Algebra, Fall 2011.
Solutions to the take-home part of Midterm #2.
1. Let F be a eld, A M atmn (F ) for some m and n, and let k = rk (A).
(a) Prove that if A = A1 + . . . + Al where rk (Ai ) = 1 for each i, then l k .
(b) Prove that there
Advanced Linear Algebra, Fall 2011. Midterm #2, take-home part.
Due Thursday, November 10th
Directions: Provide complete arguments (do not skip steps). State clearly
and FULLY any result you are referring to. Partial credit for incorrect solutions, contai
Advanced Linear Algebra, Fall 2011. Solutions to midterm #1.
1. Let V = P2 (R), the vector space of polynomials of degree 2 over R.
Let T : V V be the dierentiation map, that is, T (f (x) = f (x).
(a) Find the matrix [T ] with respect to the ordered basis
Homework #9. Due Thursday, November 3rd, in class
Reading:
1. For this homework assignment: 7.1.
2. For next weeks classes: read 7.2 and the end of 7.1, go over 7.3.
HOMEWORK POLICY: In this homework all quiz problems may be
discussed with others (followi
Homework #10. Due Thursday, November 17th, in class
Reading:
1. For this homework assignment: 7.2, 7.3.
2. For next weeks classes: read 6.8.
HOMEWORK POLICY: In this homework all quiz problems may be
discussed with others (following the previously stated
Homework #8. Due Thursday, October 27th, in class
Reading:
1. For this homework assignment: 5.2.
2. For next weeks classes: read 7.1, go over 5.4.
HOMEWORK POLICY: In this homework all quiz problems may be
discussed with others (following the previously s
Homework #1. Due Thursday, September 1st, in class
Important reminder: Only problems (or their parts) marked with the
Q (quiz) symbol need to be sumitted in writing. You are NOT allowed to
discuss those Q questions with others or use any resources (includ
Homework #6. Due Thursday, October 13th, in class
Reading:
1. For this homework assignment: 3.1, 3.2, 4.2, 4.3.
2. For next weeks class: 5.1.
HOMEWORK POLICY: In this homework all quiz problems may be
discussed with others (following the previously stated
Homework #5. Due Thursday, September 29th, in class
Reading:
1. For this homework assignment: 2.6.
2. For next weeks classes: 3.1 - 3.3. On Tuesday we will mostly talk
about the rank of matrices (3.2) and basic criteria for solvability of the linear
matri
Homework #3. Due Thursday, September 15th, in class
Reading:
1. For this homework assignment: 2.1 and the part of 2.4 dealing with
isomorphisms.
2. For next weeks classes: the rest of Chapter 2 up to 2.5 (inclusive). I
hope to send you a more detailed pla
Homework #4. Due Thursday, September 22nd, in class
Reading:
1. For this homework assignment: 2.2-2.5.
2. For next weeks classes: 2.6, 3.1 and 3.2.
HOMEWORK POLICY: The general policy is the same as in Homework#3. In this assignment there are four (4) qui
Homework #6. Solutions to selected problems.
Problem 2: In parts (a) and (b) let U, V, W be nite-dimensional vector
spaces and T : U V and S : V W linear maps.
(a) Prove that
null(ST ) null(S ) + null(T )
(recall that null(R) = dim(Ker(R) for a linear map
Homework #7. Due Thursday, October 20th, in class
Reading:
1. For this homework assignment: 5.1, parts of 5.2.
2. For next weeks class: 5.1 and 5.2.
HOMEWORK POLICY: In this homework all quiz problems may be
discussed with others (following the previously
Homework #11. Due Thursday, December 1st, in class
Reading:
1. For this homework assignment: 7.3, 6.8 and parts of 6.1.
2. For classes on Nov 29, Dec 1: read 6.8 and briey go over 6.1-6.3.
HOMEWORK POLICY: In this homework all quiz problems may be
discuss
Advanced Linear Algebra, Fall 2011. Midterm #1. Due Thursday, October 6th
Directions: Provide complete arguments (do not skip steps). State clearly
and FULLY any result you are referring to. Partial credit for incorrect solutions, containing steps in the