18.06 Problem Set 8 Solutions
Problem 1: Do problem 18 in section 6.4.
Solution
1. Suppose that A = AT and Ax = x, Ay = 0y, and = 0. Then x is in the
column space of A, and y is in the left nullspace of A since N (A) = N (AT ). But
C(A) and N (AT ) are or
(d) If the vector b is the sum of the four columns of A, write down the complete solution to
Ax = b.
Answer:
3
2
1
1
1
+ x4 2
x = + x2
0
0
1
1
0
1
2. (11 points) This problem nds the curve y = C + D 2 t which gives the best least squares t
to
18.06 Problem Set 9 Solutions
Problem 1: Do problem 15 from section 6.5.
Solution
Two ways to prove that A + B is positive denite if A and B both are.
(i) As xT Ax > 0 and xT Bx > 0 for any x, we have xT (A+B)x = xT Ax+xT Bx >
0.
(ii) As A = RT R for some
18.06 Linear Algebra, Fall 1999
Transcript Lecture 29
Okay. This is the lecture on the singular value decomposition. But everybody calls it
the SVD.
So this is the final and best factorization of a matrix.
Let me tell you what's coming. The factors will b
Exercises on factorization into A = LU
Problem 4.1: What matrix E puts A into triangular form EA = U? Multi ply by E-1 = L to factor A into LU. 1 3 0 A= 2 4 0 2 0 1 Problem 4.2: (2.6 #13. Introduction and U for the symmetric matrix a a A =
a a to Linea
18.06 Linear Algebra, Fall 1999 Transcript Lecture 4
Are we ready? Okay, ready for me to start. Ready for the taping to start in a minute. He's going to raise his hand and signal when I'm on. Just a minute, though, let them settle. Okay, guys. Okay, give
18.06 Linear Algebra, Fall 2011 Recitation Transcript LU Decomposition
BEN HARRIS: Hi. I'm Ben.
Today we are going to do an LU decomposition problem. Here's the problem right here. Find that LU decomposition of this matrix A. Now notice that this matrix A
Factorization into A = LU
One goal of todays lecture is to understand Gaussian elimination in terms of
matrices; to nd a matrix L such that A = LU . We start with some useful facts
about matrix multiplication.
Inverse of a product
The inverse of a matrix
18.06 Linear Algebra, Fall 1999 Transcript Lecture 13
OK. Uh this is the review lecture for the first part of the course, the Ax=b part of the course. And the exam will emphasize chapter three. Because those are the -0 chapter three was about the rectangu
Solving Ax = b: row reduced form R
When does Ax = b have solutions x, and how can we describe those solutions?
Solvability conditions on b
We again use the example: 1 2 2 2 8 . A= 2 4 6 3 6 8 10 The third row of A is the sum of its first and second rows,
Exercises on solving Ax = 0: pivot variables, special solutions Problem 7.1: a) Find the row reduced form of: 1 5 7 9 4 1 7 A= 0 2 -2 11 -3 b) What is the rank of this matrix? c) Find any special solutions to the equation Ax = 0. Solution: a) To transform
Exercises on solving Ax = 0: pivot variables, special solutions
Problem 7.1: a) Find the row reduced form of: 1 5 7 9 4 1 7 A= 0 2 -2 11 -3 b) What is the rank of this matrix?
c) Find any special solutions to the equation Ax = 0.
Problem 7.2: (3.3 #17.b
18.06 Linear Algebra, Fall 1999
Transcript Lecture 7
OK, here's linear algebra lecture seven.
I've been talking about vector spaces and specially the null space of a matrix and the
column space of a matrix.
What's in those spaces. Now I want to actually d
18.06 Linear Algebra, Fall 2011 Recitation Transcript Solving Ax=0
MARTINA BALAGOVIC: Hi. Welcome back.
Today's problem is about solving homogeneous linear systems, ax equals 0, but it's also an introduction to the next lecture and next recitation section
Solving Ax = 0: pivot variables, special solutions
We have a denition for the column space and the nullspace of a matrix, but
how do we compute these subspaces?
Computing the nullspace
The nullspace of a matrix A is made up of the vectors x for which Ax =
Exercises on multiplication and inverse matrices
Problem 3.1: Add AB to AC and compare with A( B + C ) : A= 1 2 3 4 B= 1 0 0 0 C= 0 0 5 6
Solution: We first add AB to AC : AB = 1 2 3 4 1 0 0 0 1 0 3 0 1 0 3 0 , AC = 10 12 20 24 1 2 3 4 0 0 5 6 10 12 20 2
Exercises on multiplication and inverse matrices
Problem 3.1: Add AB to AC and compare with A( B + C ) : A= 1 2 3 4 B= 1 0 0 0 C= 0 0 5 6
Problem 3.2: (2.5 #24. Introduction to Linear Algebra: Strang) Use GaussJordan elimination on [U I ] to find the upp
18.06 Linear Algebra, Fall 1999
Transcript Lecture 3
I've been multiplying matrices already, but certainly time for me to discuss the rules
for matrix multiplication.
And the interesting part is the many ways you can do it, and they all give the same
answ
18.06 Linear Algebra, Fall 2011
Recitation Transcript Inverse Matrices
PROFESSOR: Hi there. My name is Ana. Welcome to recitation. In lecture, you've been learning about
how to multiply matrices, and how to think about that multiplication in different way
Lecture 3: Multiplication and inverse matrices
Matrix Multiplication
We discuss four different ways of thinking about the product AB = C of two matrices. If A is an m n matrix and B is an n p matrix, then C is an m p matrix. We use cij to denote the entry
Exercises on matrix spaces; rank 1; small world graphs Problem 11.1: [Optional] (3.5 #41. Introduction to Linear Algebra: Strang) Write the 3 by 3 identity matrix as a combination of the other five permuta tion matrices. Then show that those five matrices
Exercises on matrix spaces; rank 1; small world graphs
Problem 11.1: [Optional] (3.5 #41. Introduction to Linear Algebra: Strang) Write the 3 by 3 identity matrix as a combination of the other five per mutation matrices. Then show that those five matrices
18.06 Linear Algebra, Fall 1999
Transcript Lecture 11
OK. This is linear algebra lecture eleven. And at the end of lecture ten, I was talking
about some vector spaces, but they're - the things in those vector spaces were not
what we usually call vectors.
18.06 Linear Algebra, Fall 2011
Recitation Transcript Computing the Singular Value Decomposition
PROFESSOR: Hey, we're back. Today we're going to do a singular value decomposition question. The
problem is really simple to state, find the singular value de
18.06 Linear Algebra, Fall 2011
Recitation Transcript Matrix Spaces
ANA RITA PIRES: Hi there. Welcome to recitation. In lecture, you've been learning about vector spaces
whose vectors are actually matrices or functions, and this is what our problem today
Singular value decomposition
The singular value decomposition of a matrix is usually referred to as the SVD.
This is the nal and best factorization of a matrix:
A = U V T
where U is orthogonal, is diagonal, and V is orthogonal.
In the decomoposition A = U
Exercises on similar matrices and Jordan form
Problem 28.1: (6.6 #12. Introduction to Linear Algebra: Strang) These Jordan matrices have eigenvalues 0, 0, 0, 0. They have two eigenvectors; one from each block. However, their block sizes don't match and th
Matrix spaces; rank 1; small world graphs
We've talked a lot about Rn , but we can think about vector spaces made up of any sort of "vectors" that allow addition and scalar multiplication.
New vector spaces
3 by 3 matrices We were looking at the space M o
Exercises on similar matrices and Jordan form
Problem 28.1: (6.6 #12. Introduction to Linear Algebra: Strang) These Jordan matrices have eigenvalues 0, 0, 0, 0. They have two eigenvectors; one from each block. However, their block sizes don't match and th
Exercises on independence, basis, and dimension
Problem 9.1: (3.5 #2. Introduction to Linear Algebra: Strang) Find the largest
possible number of independent vectors among:
1
1
1
1
0
0
v1 =
0
, v2 =
1
, v3 =
0
,
0
0
1
0
0
0
1
, v