May 11th , 2015
Review problems
Topics: Probability, Graph theory, Number theory, Induction, pigeon hole principle and recurrences, Combinatorics, Sets and
functions, Logic.
a
, where x R.
1 + x2
Find the value of a.
Solution: We need the pdf to integrat
ORTHOGONAL MATRICES AND THE TRANSPOSE
1. Pythagorean Theorem and Cauchy Inequality
We wish to generalize certain geometric facts from R2 to Rn .
Theorem 1.1. (Pythagorean Theorem) Given two vectors ~x, ~y Rn we have
|~x + ~y |2 = |~x|2 + |~y |2 ~x ~y = 0.
LINEAR COORDINATES
1. Bases of Rn
There are many different bases of Rn . We have already discussed the standard
basis
~e1 , . . . , ~en
where here ~ei is the vector with ith entry 1 and all others 0. If ~v1 , , ~vn is another
basis, then
S = ~v1 | | ~vn
m
Mathematics 201, Spring 2017: Assignment #3
Due: In lecture, Friday, Feb. 24
Instructions: Please ensure your name, your TAs name and your section number appear on the first
page. Also that your answers are legible and all pages are stapled. Page numbers
201 LINEAR ALGEBRA, MIDTERM 1
March 8, 2011
NAME:
Section no:
TA:
. There are 7 pages in the exam including this page.
. Write all your answers clearly. You have to show work to get points for your answers.
. Use of Calculators is not allowed during the e
GRAM-SCHMIDT ALGORITHM AND QR FACTORIZATION
1. Motivating Problem
In many situations we have a basis of a subspace, V , but want to find an orthonormal basis. This is useful, for example, in giving a formula for projV . The
Gram-Schmidt algorithm allows u
Linear algebra (math 110.201)
Final Exam - 4 May 2016
Name:
Section number/TA:
Instructions:
(1) Do not open this packet until instructed to do so.
(2) This midterm should be completed in 3 hours.
(3) Notes, the textbook, and digital devices are not permi
May 11th , 2015
Review problems
Topics: Probability, Graph theory, Number theory, Induction, pigeon hole principle and recurrences, Combinatorics, Sets and
functions, Logic.
1. The pdf of a random variable is given by
a
, where x R.
1 + x2
rn
^
Find the
Dont panic!
You have .
No calculators, books or notes allowed.
Academic Honesty Certicate. I agree to complete this exam without unauthorized
assistance from any person, materials or device.
Signature:
Date:
Name:
Section No :
(or TAs name)
Questio
Mathematics 201, Spring 2017: Assignment #2
Due: In lecture, Friday, Feb. 17
Instructions: Please ensure your name, your TAs name and your section number appear on the first
page. Also that your answers are legible and all pages are stapled. Page numbers
Mathematics 201, Spring 2017: Assignment #8
Due: In lecture, Friday, Apr. 21st
Instructions: Please ensure your name, your TAs name and your section number appear on the first
page. Also that your answers are legible and all pages are stapled. Page number
201 Linear Algebra, Practice Midterm2
Duration: 50 mins
1.
1 2 1
A= 1 0 1
0 2 1
1
0
1
Find the matrix of the transformation T (~x) = A~x with respect to the basis cfw_ 1 , 1 , 0 .
0
1
1
1 2
2. T (M ) = M
defines a linear transformation on the space of
KERNEL AND IMAGE
1. Motivating problem
Let
1
A = 1
2
2
1
3
0
1
1
we want to consider the following questions:
(1) For which ~y R3 does the system A~x = ~y have some solution?
(2) What are all solutions A~x = ~y for a fixed ~y R3 ?
First we use Gauss-Jorda
Linear Algebra Problem Set # 10
Selected Solutions
Stephen Harrop
May 6, 2017
Problem #1.
Suppose A Rnn has rank(A)=1, and let ~v im(A), ~v 6= ~0. We need to show that ~v is an
eigenvector of A. Since A has rank=1, we know that dim(im( A) = 1, so let ~a d
These are sketches of solutions just to check that you got the answers right.
(1)
The reduced row echelon form of the given matrix is
1 0 0 1 1
0 1 0 1 0
rref(A) =
0 0 1 0 1 .
0 0 0 0 0
Thus the first three columns are pivot columns and as a basis on
ORTHOGONALITY
1. Dot Product
Recall, the dot product of two vectors ~v , w
~ Rn is defined to be
v1
w1
.
.
~v = . , w
~ = . , ~v w
~ = v1 w1 + . . . + vn wn
vn
wn
The length of a vector, |~v |, is defined by
|~v | = ~v ~v .
Notice, |~v | = 0 ~v =
201 Linear Algebra, Practice Midterm Solutions
1 2 3 1
1 0 1 1
1. Row reduce the augmented matrix 3 4 7 1 to 0 1 1 1 . Therefore the solution
5 6 11 1
0 0 0 0
1 t
set can be described as vectors in R3 of the form 1 t where t R.
t
The rank of the coefficie
LEAST SQUARES SOLUTIONS
1. Orthogonal projection as closest point
The following minimizing property of orthogonal projection is very important:
Theorem 1.1. Fix a subspace V Rn and a vector ~x Rn . The orthogonal
projection projV (~x) onto V is the vector
LINEAR .ALGEBRA (MATH 110,201)
FINAL. EXAM - DECEMBER 2015
Name: _
Sect-ion mjmber/TA;
Instructions:-
(1) DO not open this packet until inst-I'LIC-i'ed to .5030.
(2) This midterm should be completed "in 3 hours,
('3) Notes, t:he.-'u-'3xtbc_>ok, and
quI-I-M If Homework. 2
gmte #319 M'fVtX cfw_S of rank 3:, [*6 has ve- -'laamj VS: gem
a Lewbnj l :3 cfw_he IAIN 19" @dj
+ha+ M '3 9f?
4&5. a dike)
Ifa coLquL con-tam: a [wolnj I,
a row;
Q5 jam more 030W". the
era-tries In that(3 aobumn are .0:
_'ma"rs*\<,
Linear algebra (math 110.201.)
Final Exam ~ 4 May 2016
Name;
Section number/TA:
Instructions:-
(1) Do not 011011 this packet 11111511 11151100th to cfw_10 5'0.
(2) This 111111101111 should be 0011111110100 111 3 hours
(_3) Notes the textbook and digit-a1-
LINEAR ALGEBRA (MATH 110.201)
FINAL EXAM - DECEMBER 2015
Name:
Section number/TA:
Instructions:
(1) Do not open this packet until instructed to do so.
(2) This midterm should be completed in 3 hours.
(3) Notes, the textbook, and digital devices are not pe
IJNEAR
ALGEBRA
gait SJtibterm ($311111.
JOHNS HOPKINS UNIVERSITY.
' SPRING 2-013
You have 50 MINUTES.
N0 calculators, books or notes allowed.
cademit Honesty Carryzcafa I agrec._to complete this Exam without unauthorized
'assistan'ee'from any pcr'so, mate