Exercises on graphs, networks, and incidence matrices
Problem 12.1: (8.2 #1. Introduction to Linear Algebra: Strang) Write down
the four by four incidence matrix A for the square graph, shown below.
(
18.06 Linear Algebra, Fall 2011
Recitation Transcript Symmetric Matrices and Positive Definiteness
PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to talk about positive definite matrices. A
18.06SC Unit 3 Exam Solutions
1 (34 pts.) (a) If a square matrix A has all n of its singular values equal to 1 in the SVD, what basic classes of matrices does A belong to ? (Singular, symmetric, ortho
18.06 Linear Algebra, Fall 2011 Recitation Transcript Graphs and Networks
NIKOLA KAMBUROV: Hi, guys. Today we're going to see how one can use linear algebra to describe graphs and networks. In particu
Symmetric matrices and positive definite ess n
Symmetric matrices are good their eigenvalues are real and each has a com plete set of orthonormal eigenvectors. Positive definite matrices are even bet
Exercises on complex matrices; fast Fourier transform
Problem 26.1: Compute the matrix F2 .
11
Solution: F2 =
, where w = ei2 /2 = 1. Hence
1w
F2 =
1
1
1 1
.
Problem 26.2: Find the matrices D and P us
Exercises on singular value decomposition
Problem 29.1: (Based on 6.7 #4. Introduction to Linear Algebra: Strang)
Verify that if we compute the singular value decomposition A = UV T of
the Fibonacci
Graphs, networks, incidence matrices
When we use linear algebra to understand physical systems, we often find more structure in the matrices and vectors than appears in the examples we make up in clas
Exercises on complex matrices; fast Fourier transform
Problem 26.1: Compute the matrix F2 .
Problem 26.2: Find the matrices D and P used in the factorization:
I
D
F2
F4 =
P
I D
F2
Hint: D is created u
p
k '~
B
'~
$
cfw_yV cfw_ycfw_ V V cfw_ ~ 'W~ S~ W%T
B
h'~ g y F cid y T cfw_y y~ b ` $ cfw_ cfw_ ` cfw_ x k R k rP ~ !y vWey 6a tytV
$ V W $
vcfw_yV ~ s V Y!G) a h g y $~ cfw_ s V a h g y 6~ cfw_
18.06 Linear Algebra, Fall 1999 Transcript Lecture 25
- one and - the lecture on symmetric matrixes. So that's the most important class of matrixes, symmetric matrixes. A equals A transpose. So the fi
Exam 3 review
The exam will cover the material through the singular value decomposition.
Linear transformations and change of basis will be covered on the nal.
The main topics on this exam are:
Eigen
18.06 Linear Algebra, Fall 2011 Recitation Transcript Pseudoinverses
DAVID SHIROKOFF: Hi everyone. Welcome back.
So today I'd like to tackle a problem on pseudoinverses. So given a matrix A, which is
Left and right inverses; pseudoinverse
Although pseudoinverses will not appear on the exam, this lecture will help us to prepare.
Two sided inverse
A 2-sided inverse of a matrix A is a matrix A-1 for
Exercises on graphs, networks, and incidence matrices
Problem 12.1: (8.2 #1. Introduction to Linear Algebra: Strang) Write down the four by four incidence matrix A for the square graph, shown below. (
18.06 Linear Algebra, Fall 1999 Transcript Lecture 32
OK, here we go with, quiz review for the third quiz that's coming on Friday. So, one key point is that the quiz covers through chapter six. Chapte
Exercises on symmetric matrices and positive definiteness Problem 25.1: (6.4 #10. Introduction to Linear Algebra: Strang) Here is a quick "proof" that the eigenvalues of all real matrices are real: Fa
18.06 Linear Algebra, Fall 2011 Recitation Transcript Exam #3 Problem Solving
DAVID SHIROKOFF: Hi, everyone. So for this problem, we're just going to take a look at computing some eigenvalues and eige
Exercises on symmetric matrices and positive deniteness
Problem 25.1: (6.4 #10. Introduction to Linear Algebra: Strang) Here is a
quick proof that the eigenvalues of all real matrices are real:
False
18.06 Linear Algebra, Fall 1999 Transcript Lecture 12
OK. This is lecture twelve. We've reached twelve lectures. And this one is more than the others about applications of linear algebra. And I'll con
Exercises on singular value decomposition
Problem 29.1: (Based on 6.7 #4. Introduction to Linear Algebra: Strang) Ver
ify that if we compute he singular value decomposition A = U V T of the
t
Fibonacc
18.06 Linear Algebra, Fall 1999
Transcript Lecture 26
Okay. This is a lecture where complex numbers come in. It's a - complex numbers
have slipped into this course because even a real matrix can have
Exercises on orthogonal matrices and Gram-Schmidt
Problem 17.1: (4.4 #10.b Introduction to Linear Algebra: Strang)
Orthonormal vectors are automatically linearly independent.
Matrix Proof: Show that Q
Exercises on Markov matrices; Fourier series
Problem 24.1: (6.4 #7. Introduction to Linear Algebra: Strang)
1b
a) Find a symmetric matrix
that has a negative eigenvalue.
b1
b) How do you know it must
18.06 Linear Algebra, Fall 2011 Recitation Transcript An Overview of Key Ideas
MARTINA BALAGOVIC: Welcome. Today's problem actually appeared in a quiz. It appeared in quiz one in fall of 1999 as quest
18.06 Linear Algebra, Fall 2011
Recitation Transcript Change of Basis
MARTINA BALAGOVIC: Hi. Welcome to recitation.
Today's this problem is about change of basis. It says the vector space of polynomia
18.06 Linear Algebra, Fall 1999
Transcript Lecture 24
- two, one and - okay. Here is a lecture on the applications of eigenvalues and, if I
can - so that will be Markov matrices. I'll tell you what a
Change of basis; image compression
Weve learned that computations can be made easier by an appropriate choice
of basis. One application of this principle is to image compression. Lecture
videos, music
18.06 Linear Algebra, Fall 2011
Recitation Transcript Markov Matrices
PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to tackle a problem in Markov matrices.
Specifically, we're going to sta
An overview of key ideas
This is an overview of linear algebra given at the start of a course on the math
ematics of engineering.
Linear algebra progresses from vectors to matrices to subspaces.
Vecto