Homework assignment 11 Section 7.3 pp. 249-250 Exercise 1. Let N1 and N2 be 3 3 nilpotent matrices over the field F . Prove
that N1 and N2 are similar if and only if they have the same minimal polynomial. Solution: If N1 and N2 are similar, they have the

Math 235: Linear Algebra HW 6
Problem 3.2.3
Problem 3.2.3
Let T be the linear operator on R3 dened by T (x1 , x2 , x3 ) = (3x1 , x1 x2 , 2x1 + x2 + x3 ) Is T invertible? If so, nd a rule for T 1 like the one which denes T . Yes it is invertible, and the i

Math 235: Linear Algebra HW 7
Problem 3.4.6
Problem 3.4.6
Let T be the linear operator on R2 dened by T (x, y ) = (y, x)
(a)
What is the matrix of T in the standard ordered basis for R2 ? 0 1 10
(a)
(b)
What is the matrix of T in the ordered basis B = cfw

Math 235: Linear Algebra HW 5
Problem 2.6.2
Problem 2.6.2
Let 1 = (1, 1, 2, 1) , 2 = (3, 0, 4, 1) , 3 = (1, 2, 5, 2) and Let = (4, 5, 9, 7) , = (3, 1, 4, 4) , = (1, 1, 0, 1)
(a)
Which of the vectors , , are in the subspace of R4 spanned by the i ? Put the

Math 235: Linear Algebra HW 4
Problem 2.3.2
Problem 2.3.2
Are the vectors 1 = (1, 1, 2, 4) , 2 = (2, 1, 5, 2) , 3 = (1, 1, 4, 0) , and (2, 1, 1, 6) linearly independent in R4 ? The vectors are not linearly independent since 1 + 2 3 4 = 0.
Problem 2.3.3
Fi

Math 235: Linear Algebra HW 1
Problem 1.6.1
Problem 1.6.1
1 2 10 Let A = 1 0 3 5 Find a row-reduced echelon matrix R which is row-equivalent to A 1 2 3 5 invertible 3x3 matrix P such that R = P A. To do this row reduce as follows: 1 2 10100 1 2 10 1 00 1

Math 235: Linear Algebra HW 1
Problem 1.2.1
Problem 1.2.1
Verify that the set of complex numbers described in Example 4 is a subeld of C Call the subeld S = x + y 2|x, y Q We must verify that S meets the following 2 conditions: 1. 0, 1 S 2. If a, b S then

MTH 235 : Linear Algebra
Sample problems - Midterm 2
1. Prove that T is a linear transformation, and nd bases for the kernel (null space) and the range of T . Is T one-to-one? Is T onto? (a) T : R3 R2 dened by T (a1 , a2 , a3 ) = (a1 a2 , 2a3 ). Solution.

MTH 235 : Linear Algebra
Sample problems - Midterm 2
1. Prove that T is a linear transformation, and nd bases for the kernel (null space) and the range of T . Is T one-to-one? Is T onto? (a) T : R3 R2 dened by T (a1 , a2 , a3 ) = (a1 a2 , 2a3 ). (b) T : R

MTH 235 : Linear Algebra
Sample problems - Midterm 1
1. (a) Argue that Z/5Z is a eld by using addition and multiplication tables. (b) Argue that Z/4Z is not a eld. Solution. (a) The addition and multiplication is inherited from that of the integers Z, so

MTH 235 : Linear Algebra
Sample problems - Midterm 1
1. (a) Argue that Z/5Z is a eld by using addition and multiplication tables. (b) Argue that Z/4Z is not a eld. 2. Let V be the vector space of functions f : R R. Show that W V is a subspace, where (a) W

Math 235: Linear Algebra HW 8
Problem 3.6.1
Problem 3.6.1
Let n be a positive integer and F a eld. Let W be the set of all vectors (x1 , . . . , xn ) in F n such that x1 + x2 + + xn = 0.
(a)
Prove that W 0 consists of all linear functaionals f of the form

pp. 73 PExercise 1.
Homework assignment 5
Which of the following maps T from R2 into R2 are linear
transformations? (a) T (x1 , x2 ) = (1 + x1 , x2 ); No, because T (0, 0) = (0, 0). (b) T (x1 , x2 ) = (x2 , x1 ); Yes, because x2 and x1 are linear homogene

Let V be the space of 2 2 matrices over F . cfw_A1 , A2 , A3 , A4 for V such that A2 = Aj for each j . j
pp. 49 Exercise 8.
Homework assignment 4
Find a basis
Solution If we start with the canonical basis for V , namely
B= B1 = 10 , B2 = 00 01 , B3 = 00

Problem Points Scores
1 15
2 13
3 4 12 5
5 5
Bonus: 10
Total: 50+10
Mat 310 Linear Algebra Fall 2004
Name:
Id. #:
Lecture #:
Test 1 (September 24 / 50 minutes)
There are 5 problems worth 50 points total and a bonus problem worth up to 10 points. Show all

Second Midterm Exam
Name: problem possible score 1 25 2 25
MAT 310, Spring 2005
ID#: 3 25 4 25 5 25 Total 100 Rec:
Directions: There are 5 problems on six pages (including this one) in this exam. Make sure that you have them all. Do all of your work in th

Homework assignment 10 Section 6.3 pp. 197-198 Exercise 1. Let V be a nite dimensional vector space. What is the minimal
polynomial for the identity operator on V ? What is the minimal polynomial for the zero operator? Solution: The minimal polynomial for

which is represented by the matrix A in the standard ordered basis for R2 , and let U be the linear operator on C2 represented by A in the standard ordered basis. Find the characteristic polynomial for T and that for U , nd the characteristic values of ea

Let V be the vector space of all polynomial functions over the eld of real numbers. Let a and b be xed real numbers and let f be the linear functional on V dened by
b
Homework assignment 8 Section 3.7 pp. 115 Exercise 2.
f (x) =
a
p(x)dx
If D is the diffe

pp. 105 Exercise 2. Let B = cfw_1, 2, 3 be the basis for C3 dened by
1 = (1, 0, 1) 2 = (1, 1, 1) 3 = (2, 2, 0). Find the dual basis of B . The rst element of the dual basis is the linear function 1 such that 1 (1 ) = 1, 1 (2 ) = 0 and 1 (3 ) = 0. To descr

Let B be the standard ordered basis for C2 and let B = cfw_1 , 2 be the ordered basis dened by 1 = (1, i), 2 = (i, 2). a What is the matrix of T relative to the pair B , B ? b What is the matrix of T relative to the pair B , B ? c What is the matrix of T

subspaces of Rn (n 3)? (a) all such that a1 0;
pp. 39-40 Exercise 1. Which of the following sets S of vectors = (a1, . . . , an) Rn are
Homework assignment 3
No. Take = (1, 0, . . . , 0) S , then () = (1, 0, . . . , n) S. / (b) all such that a1 + 3a2 = a3

Verify that the set of complex numbers of the form x + y 2, where x and y are rational, is a subeld of the eld of complex numbers. Evidently, this set contains 0 and 1. It is also easy to check that it is closed under the addition and multiplication. The

Problem Points Scores
12 6 12
3 10
4 10
5 12
Bonus: 10
Total: 50+10
Mat 310 Linear Algebra Fall 2004
Name:
Id. #:
Lecture #:
Test 2 (November 05 / 60 minutes)
There are 5 problems worth 50 points total and a bonus problem worth up to 10 points. Show all w

Solutions to Second Midterm Exam
Name: problem possible score 1 25 2 25 ID#: 3 25 4 25
MAT 310, Spring 2005
Rec: 5 25 Total 100
Directions: There are 5 problems on six pages (including this one) in this exam. Make sure that you have them all. Do all of yo

Solutions to First Midterm Exam
Name: problem possible score 1 20 2 20 ID#: 3 20
MAT 310, Spring 2005
Rec: 4 20 5 20 Total 100
Directions: There are 5 problems on ve pages in this exam. Make sure that you have them all. Do all of your work in this exam bo

First Midterm Exam
Name: problem possible score 1 20 2 20
MAT 310, Spring 2005
ID#: 3 20 4 20 5 20 Total 100 Rec:
Directions: There are 5 problems on ve pages in this exam. Make sure that you have them all. Do all of your work in this exam booklet, and cr

Problem Points Scores
1 2 3 4 5 6 12 10 10 12
Bonus: 10
Total: 50+10
Mat 310 Linear Algebra Fall 2004
Name:
Id. #:
Lecture #:
Test 2 (November 05 / 60 minutes)
There are 5 problems worth 50 points total and a bonus problem worth up to 10 points. Show all