Course Content Fall 2017
- Our goal is a careful study of basic linear algebra with an eye towards examining several applications of this important discipline. Of course, the study of applications will not be exhaustive, but hopefully will demonstrate as well the utility and importance of the underlying mathematics and its flexibility as a tool. We will discuss and emphasize the underlying mathematical principles and theorems. The basic topics are solving simultaneous linear systems (especially Gaussian elimination), matrix formulations, geometric interpretations, vector spaces, linear dependence and independence, dimension, linear transformations, orthogonality, determinants, and eigenvalues. The material is not very difficult, but we do cover a lot of material.
Basic Course Information
Math 4242 is a 4 credit course. Here is the official mathematics department description of this course:
MATH 4242 Applied Linear Algebra
Systems of linear equations, vector spaces, subspaces, bases, linear transformations, matrices, determinants, eigenvalues, canonical forms, quadratic forms, applications.
Lecture:
050 LEC , 1:25 P.M. - 2:15 P.M. , M,W,F Appleby 319 , TCEASTBANK .
Instructor: Professor S.Sperber.
Email: sperber@math.umn.edu
Office: 457 Vincent Hall
Phone: 612-625-4374
Office Hours: Mondays 2:30-3:30, Thursdays 11:30-12:20
Textbook: IMPORTANT: The selected text is ridiculously (outrageously) expensive. Please make every effort to avoid the cost of buying this text new. Other useful options: used texts are MUCH cheaper. Rental possibilities are available. Finally a PDF download seems to be available on the web.
Textbook: Gilbert Strang, Linear Algebra and its Applications, 4th edition, ISBN-13: 978-0030105678, ISBN-10: 0030105676
I believe the author has a series of video lectures which one can access from his webpage: http://www-math.mit.edu/~gs/
Organization of Course and Grading Policy
There will be two mid-semester in-class
exams of 50 minute duration. Exams will be in-class, on Wednesday, October 11 and on Wednesday, November 15.
There is also a (cumulative) two hour final examination on Monday December 18 from 8-10AM.
THE ROOM FOR THE FINAL EXAMINATION IS our current classroom Appleby 319.
Homework will be collected weekly, every Wednesday starting September 13 and some, if not all, of
your work on each problem set will be graded.This grade will be the major component of the 20% of the "homework and class participation" portion of your Final grade
The final grade will
be determined according to the following guide:
mid-semester exams 40%
final 40%
homework and class participation 20%
Sample Test
Hints and answers to sample test 2:
Problem 1a. Add C1 to C2; Add -2C1 to C3; Add -1C1 to C1; Get Row 1= (1,0,0,0) Row 2 = (0, !, !,2) Row 3 = (1,2,-1,-1) Row 4 =(0,-1,1.-1).
Add (-1)C2 to C3; Add -2C2 to C4; Get Row 1 = (1,0,0,0); Row 2 = (0,1,0,0); Row 3 = (1,2,-3, -5); Row 4 = (0,-1,2,1);
Add (-5/3)C3 to C4 Get K with row 1=(1,0,0,0), Row 2= (0,1,0,0), Row 3 = (1,2,-3,0), Row 4 = (0,-1,2, -(7/3))
So det(A) = det(K) = 7
Problem 1b. det (B^3) = det(B)^3.
det B = -6 det A
So det B = -42 and det (B^3) = (-42)^3
Problem 2.
Call basis {u,v,w}. Then c(u) = (0, -3, -1)^t, c(v) = (3,-3, -2)^t and c(w) = (4,-2,-2)^t. Find coordinates of c(u), c(v) and c(w) in terms of basis {u,v,w} In the case of
c(u), c(u) = 3u -2v -1w. Similarly c(v) = 6u -v -2w and c(w) = 6u -2w. So matrix of linear map c with respect to basis {u, v, w} has first row = (3,6,6), second row = (-2, -1, 0) and third row = (-1, -2, -2).
Problem 3. Let A^t = the 2 by 5 matrix whose first row is (-2,-1,0,1,2) and second row is (1,1,1,1,1). Let y be the column vector (1,0,3,2,2)^t. Then must solve (A^tA)(m,b)^t=
A^t y. Get. m = 2/5, b = 8/5.
Problem 4.
x_1 = 1/3, x_2 = 3/2, x_3 = -7/6
Problem 5.a.
tilde(q_1) = 1 + x. length (tilde(q_1) = sqrt(integral from -1 to 1 of (1+x)^2) = sqrt(8/3). So q_1 = sqt(3/8)(1 + x).
tilde(q_2) = b - (b, q_1) q_1. Here, I get (b, q_1) = 2sqrt 3/ 3 sqrt 2. So tilde(q_2)= (1/2) - (1/2)x -x^2. Then find length(tilde(q_2). Set q_2 = tilde(q_2)/length(tilde(q_2)).
5.b.
First row f R is ((a, q_1), (b, q_1)). Second row is (0, (b, q_2)).