Math 222 - 500
1.
Final Exam
Dec 13, 2006
(25) Define the following: a. L is a linear transformation, L is first of all a function mapping a vector space V into a vector space W with the following additional property: if and are any two vectors
Math 222 - Selected Homework Solutions for
Assignment 6
Instructor - Al Boggess
Fall 1998
Section 3.5
11 There is a mistake in this problem. It should read, the transition
matrix from F to E rather than the other way around. The transition
matrix from F =
Math 222 - Selected Homework Solutions for
Assignment 7
Instructor - Al Boggess
Fall 1998
Section 4.1
12. Let v1 ; : : : vn be a basis for V and let L1 and L2 be two linear transformations mapping V into W . We are to show that if L1 vi = L2 vi
for 1 i n
Math 222 - Selected Homework Solutions from
Chapter 4
Instructor - Al Boggess
Fall 1998
Section 4.3
11. If A and B are similar, then there is a nonsingular matrix S with
B = S ,1 AS . Therefore
det B = detS ,1 AS
= det S ,1 det Adet S
where we have used
Math 222 - Selected Homework Solutions from
Chapter 5.2
Instructor - Al Boggess
Fall 1998
Section 5.2
1d The given matrix is
2
3
4 ,2
61
7
6
A := 6 2 3 7
4
5
17
34
The two column vectors are clearly linearly independent one is not a
scalar multiple of the
Math 222 - Selected Homework Solutions from
Chapter 5.3
Instructor - Al Boggess
Fall 1998
Section 5.3
6. To show the second property, we have
Zb
hf; gi =
a
Zb
f xgx dx
=
gxf x dx
a
= hg; f i
To show the third property, we have
h f + g; hi =
Zb
a
f + gh d
Math 222 - Selected Homework Solutions for
Assignment 2
Instructor - Al Boggess
Spring 1998
Page 76 - 1.5
14 We are given that A is an n n matrix with Ax = 0 for all vectors x.
We are to show that A = 0 which means that we must show the ith,
jth entry, a
Math 222 - Selected Homework Solutions from
Chapter 6.1
Instructor - Al Boggess
Fall 1998
Section 6.1
4. If A is a nonsingular matrix with eigenvalue , then Av = v for some
nonzero vector v. Applying A,1 to both sides of this equation, we
obtain
v = A,1Av
Math 222 - Selected Homework Solutions from
Chapter 5.5, 5.6.
Instructor - Al Boggess
Fall 1998
Section 5
10. Prove that the transpose of an orthogonal matrix is orthogonal. If Q
is orthogonal, then Qt Q = I . Therefore Qt is the inverse of Q. To
show tha
Math 222 - Selected Homework Solutions for
Assignment 5
Instructor - Al Boggess
Fall 1998
Section 3.3
14 We are given that A is an m n matrix with linearly independent
columns a1 ; : : : a . We are to show the null space equals 0. This
means we must show
Math 222 - Selected Homework Solutions from
Sections 3.1 and 3.2
Instructor - Al Boggess
Fall 1998
xercises for Section 3.1
E
9 a Show 0 = 0 for each scalar . We must show 0 + x = x for all
vectors x. For then, 0 must be 0 since the Zero in the vector spa
Math 222-500
1.
Solutions Exam 1
October 10, 2006
(15) Define the following: a. the span of the set of vectors 1 , 2 , , k , x x x The set of all linear combinations of these vectors is the span.
b.
the set of vectors 1 , 2 , , k is linearly inde
Math 222 - 200 1. (20) Define the following terms:
Exam 1 Solutions
February 25, 2005
(a) linearly independent set of vectors A set of vectors {x1 , , xk } is linearly independent if whenever 1 x1 + + k xk = 0, then 1 = 2 = = k = 0. (b) s
Math 222-500
1.
Solutions Exam 2
November 21, 2006
(15) Define the following: a. coordinates of a vector, If V is a vector space, and S 1 , , n is a basis of V, then the u u coordinates of a vector V with respect to the basis S are the unique
Math 222 - 200
Solutions Exam 2
April 1, 2005
1. (30) Let T : V V be a linear transformation. Let {v1 , v2 , v3 } be a basis of V . Suppose the matrix representation A of T with respect to this basis is 1 2 0 A= 0 1 -4 . -2 -5 0 (a) In terms o
Math 222 - 200
Solutions Final Exam
April 27, 2005
1. (15) What shall I name my pond snake? The top three names are: Eigen from Ashley Hubble Pond, James Pond agent 008 from Amy Hopson Sam, aka `Snake at Mike's' from Valerie Berlin
2. (25) Let T
Chapter 16 Numerical Linear Algebra
16.1 Sets of Linear Equations
MATLAB was developed to handle problems involving matrices and vectors in an efficient way. One of the most basic problems of this type involves the solution of a system of linear equa
Math 222 - Selected Homework Solutions from
Sections 2.2 and 2.3
Instructor - Al Boggess
Fall 1998
Page 98 - Section 2.2
5 We are to show det A = n det A. The key is to write
A = I A
Now take determinants:
det A = det IA = det I det A
Now, I has n factors
Math 222 - Selected Homework Solutions
Instructor - Al Boggess
Spring 1998
Page 28
12 The equations are found by equating the tra c owing into each node
with the tra c owing out. Going counterclockwise starting with the
node in the upper left, we get the