Math 545
Linear transformations and the geometry of surfaces
A homework assignment
Let S be a smooth surface in R3 given by the equation f (x, y, z ) = 0, where smoothness means that the gradient vector
f :=
f f f
,
,
x y z
does not vanish at any point o
Gram-Schmidt example from class
I didnt have time to finish the example for the Gram-Schmidt process. Sorry for the lateness,
but here it is for the sake of completion. It is an example from Sean Uppals notes.
Example: Consider V = P2 (R), with inner prod
CHAPTER
6
Woman teaching
geometry, from a
fourteenth-century
edition of Euclids
geometry book.
Inner Product Spaces
In making the definition of a vector space, we generalized the linear structure
(addition and scalar multiplication) of R2 and R3 . We igno
Elementary Linear Algebra
Kuttler
September 5, 2013
2
CONTENTS
1 Some Prerequisite Topics
1.1 Sets And Set Notation . . . . . . .
1.2 Well Ordering And Induction . . .
1.3 The Complex Numbers . . . . . . .
1.4 Polar Form Of Complex Numbers .
1.5 Roots Of
Solutions
Math 545 Exam # 1
October 6, 2016
1. (16 points) Complete the following definitions:
(a) Let A and B be n n matrices. Then A is similar to B if and only if there exists an n n matrix
M such that B = M 1 A M .
(b) An n n matrix Q is orthogonal if
MATH 545 Fall 2016
HW # 2 Due: Friday, September 23
Solutions
Problem 1. In class we stated the following Theorem:
Let V be a finite-dimensional vector space and B a finite subset of V .
Then the following three conditions are equivalent:
a) B is a basis
Elementary Linear Algebra
Kuttler
September 5, 2013
2
CONTENTS
1 Some Prerequisite Topics
1.1 Sets And Set Notation . . . . . . .
1.2 Well Ordering And Induction . . .
1.3 The Complex Numbers . . . . . . .
1.4 Polar Form Of Complex Numbers .
1.5 Roots Of
A= [1,2;3,4]
B=[77;%pi]
/solve Ax=b
x=A\B
/ Lets build an eigenvalue problem
lama=[0;2;1];
D=diag(lama);
/Diagonal matrix with lama on the diagonal
v1=[1;1;0];
v2=[0;1;-1];
v3=[3;0;4];
s=[v1,v2,v3];
/ Aeig=S*D*(inv(s); or better
Aeig=s*D/s;
/?:\? INVERSE
Study Guide
Any topics covered in the course are fair game for the exam. The following is a general guide to
the topics you are expected to know, but is not a strict limit on what may be asked.
(1) Be able to define the following terms and concepts:
Orth
A=zeros(100,100)
for (i=1:99)
for (j=1:99)
if(i=j) A(i,j)=2;A(i+1,j)=-1;A(i,j+1)=-1; end
end
end
A(100,100)=2
b=zeros(100,1)
b(1,1)=1
y=A\b
x=zeros(100,1)
xs=zeros(100,1)
disp("Jacobi")
D=zeros(100,100)
L=zeros(100,100)
U=zeros(100,100)
for (i=1:99)
for (
A=zeros(100,100)
for (i=1:99)
for (j=1:99)
if(i=j) A(i,j)=2;A(i+1,j)=-1;A(i,j+1)=-1; end
end
end
A(100,100)=2
b=zeros(100,1)
b(1,1)=1
y=A\b
x=zeros(100,1)
D=zeros(100,100)
L=zeros(100,100)
U=zeros(100,100)
for (i=1:99)
for (j=1:99)
if(i=j) D(i,j)=2;L(i+1,
Homework 5: Due April 1st, 2016
1. Permutations
(a) Write down all possible 3 by 3 permutation matrices.
(b) Give an example of distinct permutation matrices P1 and P2 such
that P1 P2 = P2 P1 . Give an example of distinct permutation matrices
P3 and P4 su
Homework 2: Due Feb. 10th, 2016
All answers must be justified.
1. Let V = C 1 ([0, 1]; R) denote the vector space of continuous functions
whose derivatives are continuous functions on the closed interval [0, 1].
Prove whether or not the following defines
Math 545
Extra problem on the Primary Decomposition Theorem
2 10
Let A be the matrix 0 2 0 . In this problem you are asked to work out
3 1 5
explicitly for the matrix A the construction that is carried out in the proof of the
Primary Decomposition Theorem
Math 545
Solution of Midterm 1
Fall 2010
1. (20 points) Let U , V , and W be vector spaces of dimensions m, n, and p, respectively. Let S : U V and T : V W be linear transformations satisfying T S = 0
(the zero linear transformation from U to W ). Prove t
Math 545
Midterm 1
Fall 2010
Name:
1. (20 points) Let U , V , and W be vector spaces of dimensions m, n, and p, respectively. Let S : U V and T : V W be linear transformations satisfying T S = 0
(the zero linear transformation from U to W ). Prove that ra
Solution to Homework Assignment 1
Qian-Yong Chen
Section 1.2 (Page 9):
#8: Operations of (Equation 1 + Equation 3 - Equation 2) will lead to 1 = 0. Replace the RHS
of the third equation by 1, the solution exists. One such solution is u = 3, v = 1, w = 0.
Solution to Homework Assignment 2
Qian-Yong Chen
Section 1.5 (Page 39):
#10: (a) Since L is a lower triangular matrix, and U is an upper triangular matrix.
(b) It takes approximately n3 /3 operations to nd the LU decomposition, and then n2 /2+ n2 /2
to so
Solution to Homework Assignment 3
Qian-Yong Chen
Section 2.1 (Page 73):
#24: For this type of problems, the rst thing is to reduce the matrix to Echelon form by row
operations. Then check the rows of the coefcient matrix with all zeros. The solution exist
Solution to Homework Assignment 4
Qian-Yong Chen
Section 2.4 (Page 110):
#14: For a matrix of size m-by-n, the right inverse exists if and only if m n and its rank is
equal to m. The left inverse exists if and only if n m and its rank equals n.
So for mat
Solution to Homework Assignment 5
Qian-Yong Chen
Section 3.1 (Page 148):
#4: The product of the i-th row of B and the j -th column of B 1 is just the (i, j ) element of
B B 1 = I , i.e., 1 if i = j , zero otherwise.
#22: Rewrite x1 + x2 + x3 + x4 = 0 as
1
Solution to Homework Assignment 6
Qian-Yong Chen
Section 3.3 (Page 170):
#10: Assume A = (a1 , a2 ), then AT A is the 2-by-2 identity matrix, and AT b =
0
. The
0
projection of b onto the plane of a1 and a2 is the zero vector.
#12:
(a) For any vector (x1
Homework 1: Due Feb 1st, 2016
This homework contains review material, with references to page numbers
in the Kuttler text.
1. Define the matrices
A=
1 2
1 1
1 0
B= 4 1
2 1
8
1
3
1
0
Compute AB and BA. If a product does not exist, state why.
2. (a) Put the
Homework 6: Due April 15th, 2016
1. Let A R22 . Show that the Jacobi iteration converges for all starting
guesses if and only if the Gauss-Seidel iteration converges for all starting
guesses.
2. Computing (For this problem, you do not need to show work by
Homework 4: Due Mar. 11th, 2016
1. Markov Matrices Show that all eigenvalues of a Markov Matrix A
satisfy | 1. (Hint: for any eigenvector v of AT , we can scale so that
maxi |vi | = 1. Show that |(AT v)i | 1.)
2. Convergence of Markov Matrices Suppose A i
Homework 3: Due Feb 26th, 2016
1. Application: Choosing a Norm For vectors v Rn , define the norms
kvk = max |vj |
j=1,.,n
and
kvk1 =
n
X
|vj |.
j=1
The Euclidean norm, that is, the norm induced from the standard inner
product is written kvk2 .
(a) First,