Solutions to problem set #1
ad problem 1: Solution to Section 1.2, problem 13.
Several answers are possible. For example, (1, 0, 1) and (0, 1, 0) t the bill.
Remarks: A more systematic approach is given by the Gram-Schmidt process (4.4). But in this speci
Problem set 6 solutions
Question 1
To nd one vector in the plane, we set x = 1 and z = 0. This forces y = 1 and we get (1, 1, 0).
Any vector orthogonal to this vector has x = y so set x = y = 1. The plane equation forces z = 1
and we (1, 1, 1). (1, 1, 0)
18.06 Problem Set 7
Solutions
Problem 1. Section 5.2, Problem 3, page 263. Clarication. In this problem xs
symbolize 5 nonzero entries (not necessarily equal).
Each cofactor Cij is given by: Cij = (1)i+j det(Mij ), where the submatrix Mij
is obtained from
10/17/2014
Notebook
The purpose of this assignment is to show that one can obtain the Legendre Polynomials numerically
from the QR decompostion
The Legendre polynomials are orthogonal on [-1,1] and have many applications. Please see
http:/en.wikipedia.org
18.06 Problem Set 7
due Thursday, October 30, 2014, before 4:00 pm (sharp deadline) in Room E17-131
Write down all details of your solutions, not just the answers. Show your reasoning. Please staple the pages together and clearly write your name, your
rec
10/10/2014
Notebook
This problem is about predicting the world population.
Form the matrix of data for the world population from http:/en.wikipedia.org/wiki/World_population
(http:/en.wikipedia.org/wiki/World_population), table "Estimated world and region
18.06 Problem Set 8
due Thursday, November 13, 2014, before 4:00 pm (sharp deadline) in Room E17-131
Write down all details of your solutions, not just the answers. Show your reasoning. Please staple the pages together and clearly write your name, your
re
PS7
1. We would like to you play with the famous quarter circle law which states that if you draw a
1
scaled histogram of the eigenvalues of a random matrix n AAT , where A is normally distributed, the
picture if scaled correctly looks like a quarter circ
18.06 Problem Set 8. Solutions
Problem 1. Section 6.2, Problem 14, page 309.
The matrix A = [ 3 1 ] is not diagonalizable because the rank of A 3I is
.
03
Change one entry to make A diagonalizable. Which entries could you change?
Solution. The rank of A 3
18.06 Computational PSet 4
You may use any computer language. Please submit printouts with the problem set.
1. This problem asks you to solve for voltages in currents following the methodology of Example 1
on page 427.
Set up the incidence matrix for the
18.06 Problem Set 9
due Thursday, November 20, 2014, before 4:00 pm (sharp deadline) in Room E17-131
Please note that the problems listed below are out of the 4th edition of the
textbook. Please make sure to check that you are doing the correct problems.
Problem Set 9 Solutions
Question 1: Section 6.4, Problem 4, page 338
2 1
5 0
Let Q =
and =
. Then Q is orthogonal and A = QQT .
1 2
0 10
One could also swap the columns of Q and their sign. If one swaps the columns, one must swap
5 and 10.
1
5
Question 2:
Solutions to problem set #10
ad problem 1: (a) The required SVD is A T = V T U T .
(Proof: In fact, A = UV T leads to A T = UV T
T
= VT
T
T U T = V T U T ,
=V
and each of V and U is a matrix with orthonormal columns.)
Dont forget the transposition in T !
18.06 Problem Set 10
due Monday, December 01, 2014, before 4:00 pm (sharp deadline) in Room E17-131
The due date is extended till Monday because of Thanksgiving vacation.
Problem 1. Let A be a matrix with SVD A = U V T .
(a) Find an SVD of AT .
(b) Find a
Linear Algebra
Jim Hefferon
1 3
2 1
1 3
2 1
x1 1 3
2 1
x1 x3
2 1
6 8
2 1
6 8
2 1
Notation R N C cfw_. . . . . . . V, W, U v, w 0, 0V B, D En = e1 , . . . , en , RepB (v ) Pn Mnm [S ] M N V W = h, g H, G t, s T, S RepB,D (h) hi,j |T | R (h), N (h) R (h), N
Linear Algebra in Twenty Five Lectures
Tom Denton and Andrew Waldron
March 27, 2012
Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw
1
Contents
1 What is Linear Algebra?
12
2 Gaussian Elimination
19
2.1 Notation for Linear Systems . . . . . . .
18.06 Problem Set 6
due Thursday, October 23, 2014, before 4:00 pm (sharp deadline) in Room E17-131
Write down all details of your solutions, not just the answers. Show your reasoning. Please staple the pages together and clearly write your name, your
rec
PS5 Solution
This problem is about predicting the world population. Here is a julia notebook. You can convert
to any language. Form the matrix of data for the world population from
http:/en.wikipedia.org/wiki/World_population, table Estimated world and re
18.06 Problem Set 4. Solutions
Problem 1. Section 3.5, Problem 16, page 180.
Find a basis for each of these subspaces of R4 .
All vectors whose components are equal.
All vectors whose components add to zero.
All vectors that are perpendicular to (1, 1,
Exam Solutions
Problem 1
1
0
1
(a) Do Gram-Schmidt orthogonalization for the vectors a1 = 0, a2 = 1, a3 = 2.
1
0
3
(b) Find the A = QR decomposition for the matrix
0 1
.
1 2
(c) Find the projection of the vector (1, 0, 0)T onto the line spanned by t
Exam Solutions
Problem 1
1 1 1
Let A = 2 4 4 .
3 7 10
(a) Find the A = LU factorization of the matrix A.
(b) Solve the
1
2
Let B =
3
system Ax = (3, 10, 20)T .
1 1
4 4 (obtained by replacing the bottom right entry by the parameter k).
7 k
(c) For which va
18.06 Problem Set 1
due Thursday, September 11, 2014, before 4:00 pm (sharp deadline) in Room E17-131
Write down all details of your solutions, not just the answers. Show your reasoning. Please staple the pages together and clearly write your name, your
r
PS1 Solution Template
a. Using the matrix squaring operator create a triangular matrix with 1 on the main diagonal, 2
above, etc.
1 2 . n 1
n
1 2
.
n1
.
.
M (n) =
.
.
1
2
1
In[1]:
Out[1]:
# The line below is a Julia function that given n computes M(n)
18.06 Computational PSet 1
You may use any computer language. We encourage trying out Julia. Please
submit printouts with the problem set.
a.) Create the n n matrix
1 1 . 1 1
1 1 . 1
. .
En =
.
.
.
1 1
1
In some languages this is triu(ones(n,n).
Using
PS1 Solution Template
a. Using the matrix squaring operator create a triangular matrix with 1 on the main diagonal, 2
above, etc.
1 2 . n 1
n
1 2
.
n1
.
.
M (n) =
.
.
1
2
1
In[1]:
Out[1]:
In[2]:
Out[2]:
# The line below is a Julia function that given n
If you use the codes here, you should still do two important things:
1) make sure you know how the code works and
2) explain your answer as if you did not have a computer
Pascal's Triangle
I [] fnto An
n 1: ucin ()
#Psa Mti
acl arx
pzrsItnn #Itmyb oitdt w
18.06 Computational PSet 2 solutions
In high school you may have learned one way to multiply matrices. Chatper 2.4 of your text illustrates
many ways.
1. The high school or dot product way appears on pages 67 and pages 68. It says that the entry in
the (i
18.06 PROBLEM SET #2 - SOLUTIONS
FALL 2014
1. Section 2.6, Problem 13, page 104.
The
four conditions are a = 0, b = a, c = b, d =
c.
a a
a
a
1000
0 ba ba ba
1 1 0 0
U =
L=
0 0 cb cb
1 1 1 0
0 0
0 dc
1111
2. Section 2.6, Problem 16, page 105.
Solving L