LINEAR ALGEBRA SAMPLE MIDTERM ANSWERS
(1) Circle the correct answer True/False for each of the following:
T If A is a matrix then Ker(A) is the set of solutions of the linear system
Ax = 0
F The set of solutions of a linear system Ax = b is always a linea
LINEAR ALGEBRA SAMPLE MIDTERM 2
Linear space will always means real linear space (i.e., the scalars are R) and
matrices have real entries unless explicitely stated.
(1) Circle the correct answer True/False for each of the following:
F If a linear space ha
LINEAR ALGEBRA SAMPLE MIDTERM 2
Linear space will always means real linear space (i.e., the scalars are R) and
matrices have real entries unless explicitely stated.
(1) Circle the correct answer True/False for each of the following:
T/F If a linear space
LINEAR ALGEBRA SAMPLE MIDTERM
(1) Circle the correct answer True/False for each of the following:
T/F If A is a matrix then Ker(A) is the set of solutions of the linear system
Ax = 0
T/F The set of solutions of a linear system Ax = b is always a linear
su
Linear Algebra HW # 13
Woo Chang Chung
December 16, 2011
Section 9.1
Coupled linear ordinary dierential equations can be solved using the technique described in pg. 404. You
will nd in each problem that you have two distinct eigenvalues, and two is the nu
HW #11 partial solution
Woo Chang Chung
December 7, 2011
6.3
Some of you need to practice calculating determinants.
26
A 1 =
adj (A)
adj (A)
=
= adj (A)
det(A)
1
Since i, j entry of adj (A) is (1)i+j det(Aji ), each entry of adj (A) is an integer, because
Linear Algebra HW # 13
Woo Chang Chung
December 16, 2011
Section 9.1
Coupled linear ordinary dierential equations can be solved using the technique described in pg. 404. You
will nd in each problem that you have two distinct eigenvalues, and two is the nu
LINEAR ALGEBRA HOMEOWORK 13, DUE MONDAY DEC 12
This homework is due exceptionally on MONDAY
Problems:
Section 9.1, problems 26, 28, 30
Section 9.2 Problems 30, 32
1
LINEAR ALGEBRA HOMEOWORK 12 ANSWERS
Section 7.3 Problems 24, 38, 52
2
1
24. You need a matrix with eigenvalues 1,1 and eigenvector
. One calculates
a
that this forces the matrix to be of the form (a a1)/2 22 2a for some a. For a = 1
it has two-dimensional
LINEAR ALGEBRA HOMEOWORK 12, DUE WED DEC 7
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 7.3 Problems 24, 38, 52
Section 7.4 Problems 2
SOME EXTRA ANSWERS TO LINEAR ALGEBRA
HOMEOWORK 11
Woo Chang Chung has kindly provided solutions. I am adding a few more.
Section 6.3 Problems 24, 26, 30, 34, 44
2
80
3 5
6 1 7
=
24. x = 1, y = 2, z = 1. For example: y =
230
det0 4 5
607
T
18
00
18 12 2
1
LINEAR ALGEBRA HOMEOWORK 11, DUE WED NOV 30
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 6.3 Problems 24, 26, 30, 34, 44
Section 7.1 P
LINEAR ALGEBRA HOMEOWORK 10, DUE WED NOV 23
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 6.1 Problems 8, 10, 16, 26, 30, 34, 44
Sectio
V2010 HW # 9
Woo Chang Chung
November 11, 2011
Book 5.5.10
11
t f (t) dt = 0. In other words,
2 1
we rst nd the basis of the orthogonal complement of Span(g (t) = t).
We want to nd polynomials f (t) of degree at most 2 such that
1
2
1
t at2 + bt + c dt =
LINEAR ALGEBRA HOMEWORK 8 ANSWERS
Section 5.3 Problems 2, 4, 6, 10, 36, 38, 42, 54
2. Yes, columns are orthonormal.
4. No, rst and last columns are not orthogonal.
6,10. Yes for both. E.g., use the facts that that In is orthogonal, a product of
orthogonal
LINEAR ALGEBRA HOMEOWORK 8, DUE WED NOV 2
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 5.3 Problems 2, 4, 6, 10, 36, 38, 42, 54
Sectio
LINEAR ALGEBRA HOMEOWORK 7, DUE WED OCT 26
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 5.1 Problems 12, 14, 16, 26, 28 (you may want
LINEAR ALGEBRA HOMEOWORK 6, DUE WED OCT 19
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 4.1 problems 16, 20, 32, 53, 54, 57
Section 4.
V2010 HW #5 Solution
Woo Chang Chung
November 5, 2011
3.4.8
12
2
10
6
1 0 c1 = 3 1 0 c1 = 3 c1 = 6 and c1 = 3 (contradiction)
c2
c2
01
4
01
4
No, x is not in the span of v1 and v2 .
3.4.10
1
2
5
3v1 + v2 = 3 0 + 1 = 1
1
0
3
Yes, x is in the span of v1 an
LINEAR ALGEBRA HOMEOWORK 5, DUE WED OCT 12
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 3.4 problems 8, 10, 20, 22, 38, 44, 50, 60, 65
LINEAR ALGEBRA HOMEOWORK 4, DUE WED OCT 5
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Reminder: First Midterm is this coming Wednesday October 5. It co
SELECTED ANSWERS HOMEWORK 3
Section 2.3 problem 34. A is reection across the xaxis. A2 = I2 , A3 = A,
A = I2 , . . . , A1000 = I2 , A1001 = A,
4
Section 2.3 problem 36. A is a shear map. An =
1
0
n
1
.
Section 2.3 problem 44. One can choose A to be rotati
LINEAR ALGEBRA HOMEOWORK 3, DUE WED SEP 28
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Section 2.3 problems 10, 14, 16, 20, 34, 36, 44
Sectio
ANSWERS TO SELECTED PROBLEMS HOMEWORK 2
Section 1.3 problem 24: Let A be a 4 4 matrix and let b and c be two
vectors in R4 . The system Ax = b has a unique solution. What can you say about
the number of solutions of the system Ax = c
Solution: Since Ax =
LINEAR ALGEBRA HOMEOWORK 2, DUE WED SEP 21
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
We did not cover shears (pp.6364) in class but you will need to
LINEAR ALGEBRA HOMEOWORK 1, DUE WED SEP 14
Homework should be in the homework box opposite room 410 Math (Mathematics
Building, Columbia Campus) before the beginning of class each Wednesday.
Problems:
Pages 58, problems 10,20, 40, 48. (48 is really as muc
COMPLEX LINEAR SPACES AND HERMITIAN INNER
PRODUCTS
These are brief notes on what we covered in class on linear spaces over the
complex numbers and hermitian linear spaces.
We denote the set of complex numbers a + ib by C. Recall that i here denotes
1 and