MA 590
Homework 10
December 3, 2013
1. Consider the linear functional on R3 dened by (x) = x1 + 2x2 + 3x3 for
x = (x1 , x2 , x3 )T R3 .
a. (4 points) Find w R3 such that (x) = x, w for all x R3 .
b. (6 points) Determine the adjoint transformation : R1 R3
MA 590
Final Exam Solutions
December 16, 2013
In working the exam, you may use any notes and, if desired, one book (but no more than
one). Calculators are also allowed but shouldnt be necessary. Show your work on all but
the rst problem. Use the backs of
MA 590
Mid-Term Exam
October 15, 2013
In working the exam, you may use any notes and, if desired, one book (but no more than one).
Calculators are also allowed but shouldnt be necessary. Show your work on all but the rst problem.
Use the backs of the page
University, Faculty:
University of Preov in Preov
Faculty of Humanities and Natural Sciences
Department of Physics, Mathematics and Techniques
COURSE DESCRIPTION
Code:
2MAT/LMATH/12
Title: LINEAR ALGEBRA
Field of study: 1. 1. 1 Teaching of general educati
Linear Algebra
4/20/05
Using diagonalization
to compute powers, inverses, roots, and exponentials
If you can diagonalize a matrix, then you can compute its powers efficiently. You can
also compute its roots, inverse, and exponential.
Suppose A is nn, and
Linear Algebra
Department of Economics
Spring 2016
National Taipei University
Class time: 14:10-17:00 Thursday
Classroom: room 1F13
Instructor: Chien-Ho Wang
Office: room 3F36
Office phone number: (02)86741111-67172
e-mail: [email protected]
Offic
LINEAR ALGEBRA FINAL EXAM.
MARCH 2010
YOUR NAME:
Duration: 3 hours.
Please answer all questions. Points for each parts are in ( ).
Explain your solutions, quote theorems you are using.
1. Let T : R 4 [ x ] R 32 be the linear transformation defined by:
c 3
Using Technology to Teach Linear Algebra
Lawrence E. Levine
Professor
Department of Mathematical Sciences
Stevens Institute of Technology
Hoboken, NJ 07030
201-216-5425
201-216-8321 (FAX)
[email protected]
http:/attila.stevens-tech.edu/~llevine/
Co
William Paterson University of New Jersey
College of Science and Health
Department of Mathematics
Course Outline
1.
2.
3.
4.
5.
6.
Title of Course, Course Number and Credits:
Linear Algebra- Math 2020
3 credits
Description of Course:
An introductory cours
Linear Algebra Quiz
This is a graded examination.
Take this exam when and where you like.
Take no more than two hours.
Do not use a calculator; computer; text, notes, or any other reference material; or any
consultation. You may use scratch paper.
You may
COLLEGEWIDE COURSE OUTLINE OF RECORD
MATH 265, LINEAR ALGEBRA
COURSE TITLE: Linear Algebra
COURSE NUMBER: MATH 265
PREREQUISITES: MATH 212 Calculus II
SCHOOL: Liberal Arts and Sciences
PROGRAM: Liberal Arts
CREDIT HOURS: 3
CONTACT HOURS: Lecture: 3
DATE O
COMS 4721: Machine Learning for Data Science
Course Information
1/17/2017
Prof. John Paisley
Columbia University
C OURSE I NFORMATION
Instructor: Prof. John Paisley
Email: [email protected]
Office Hour: Monday 11-12, Mudd 422
Course Website:
http:/www
MA 590
Homework 1
September 3, 2013
1. a. (10 points) Show that if X and Y are subspaces of a vector space V , then X Y
is also a subspace of V .
b. (5 points) Consider
x1
X = x x2 I 3 : a 1 x1 + a 2 x2 + a 3 x3 = 0 ,
R
x3
x1
x2 I 3 : b1 x1 + b2 x2 + b3
MA 590
Homework 6
October 29, 2013
You are not required to hand in this assignment.
The rst two problems are taken from problems 1 and 5, page 427, of the Strang reference.
1. For the following A, nd M and J for the Jordan form A = M JM 1 .
012
11
a. A =
MA 590
Homework 3
September 17, 2013
1. (10 points) In each of the following, describe the null-space and range of the given linear
transformation and say whether it is 1-1 or onto (or both).1
a. T : R2 R2 given by T (x) = uuT x, where u =
1
2
1
.
1
b. T
MA 590
Homework 5
October 8, 2013
1. (10 points) The trace of a matrix A Rnn (with entries aij ) is dened by trace A =
n
i=1 aii . Show that the trace is invariant under similarity transformations, i.e., trace A =
trace M AM 1 for nonsingular M Rnn .
Sugg
MA 590
Homework 2
September 10, 2013
1. (10 points) Consider the functions j C [0, 1] for j = 1, 2, . . . , dened by
j (x) = sin jx for 0 x 1. Show that any nite subset cfw_1 , . . . , n is linearly
independent.
Hint: You can easily verify (but no need t
MA 590
Homework 8
November 12, 2013
1. (5 points) The unit sphere in a vector space V with norm is cfw_v V : v = 1.
Sketch the unit sphere in R2 for the norms 1 , 2 , and introduced in class.
2. (10 points) Suppose that V is a vector space with inner prod
MA 590
Homework 11 (with Solutions) December 10, 2013
Needless to say, this is not to be handed in.
1. a. (5 points) Find a Householder transformation H = I 2uuT such that Hv = w,
where v = (2, 1, 2)T and w = (3, 0, 0)T .
1
2
1
Solution: H = I 2uuT , wher
MA 590
Homework 10 Solutions
December 3, 2013
1. Consider the linear functional on R3 dened by (x) = x1 + 2x2 + 3x3 for
x = (x1 , x2 , x3 )T R3 .
a. (4 points) Find w R3 such that (x) = x, w for all x R3 .
The vector is w = (1, 2, 3)T . I should have ment
MA 590
Homework 4
October 1, 2013
1. (10 points) Suppose A R33 is such that det A = 3. Find det 2A, det A, det A2 , and
det A1 .
2. (5 points) A matrix U Rnn is said to be orthogonal if U T U = I .1 What are the
possible values of det U ?
3. (5 points) A
MA 590
Homework 7
November 5, 2013
1
1. (10 points) Preamble. The function f (x) = 1x is analytic for |x| < 1. Indeed, its
Taylor series is just the geometric series f (x) = xk , which converges absolutely for
k=0
|x| < 1. The Spectral Mapping Theorem tel
MA 590
Homework 6
October 29, 2013
You are not required to hand in this assignment.
The rst two problems are taken from problems 1 and 5, page 427, of the Strang reference.
1. For the following A, nd M and J for the Jordan form A = M JM 1 .
012
11
a. A =
MA 590
Homework 9
November 19, 2013
n
1. (10 points) You saw in Homework 8 that the functions
2
0
sin kx
k=1
are orthonormal
in the inner product f , g =
f (x)g (x) dx and, therefore, constitute an orthonormal basis
of Sn = cfw_f C [0, ] : f (x) = n=1 k s
MA 590
Homework 9 Solutions
1. (10 points) You saw in Homework 8 that the functions
November 19, 2013
2
0
n
sin kx
k=1
are orthonormal
in the inner product f , g =
f (x)g (x) dx and, therefore, constitute an orthonormal basis
of Sn = cfw_f C [0, ] : f (x)
Algorithms for Data Science
CSOR W4246
Eleni Drinea
Computer Science Department
Columbia University
Eigen-analysis and SVD for PCA
Today
1 Review or eigenvalues and eigenvectors
2 Application I: analysis of Hubs & Authorities and PageRank
3 High-dimension