LINEAR ALGEBRA SPRING 2014 UPDATE PAGE
SITARAMAN

Welcome.
Here you will see info about what we did in class, upcoming exams, solutions for past exams, etc.,
CHECK AT LEAST ONCE A WEEK!!

OLD FINAL EXAMS

CELL-PHONE AND LAPTOP USE IN CLASS PROHIBITED UNLESS REQUIRED BY INSTRUCTOR

WHAT IS LEARNING MATH ALL ABOUT?

FREE Howard based online homework website for PRACTICE ONLY : WEBWORK
Use practice1 (or practice2 or practice3...) as username and the same as password.

TUTORIAL LAB M-F 8.30am-6pm (fri upto 5pm) in ASB-B room 109

EXAMS WILL BE ANNOUNCED BELOW

NOTE: CLASSROOM HAS BEEN MOVED TO CHB 306

Final Exam Solutions
Solutions and Comments for Final exam
Updated 4/30/2014

Final Exam Wed 30 April 10.30-12.30pm CHB 306
Review Monday April 28 in ASB-B 213 from 2 to 4 pm.
Solutions for final exam from spring 2011.
Brief Solutions for final exam for graduating seniors from spring 2011.
OTHER OLD FINAL EXAMS.
Also study class notes, solutions to quizzes and tests below and those from spring 20111 and 2009 (see above), and as many review problems listed below as possible.
Solutions and Comments on Quiz 17 from today
Today we talked about 5.3 (diagonalization) specifically how to diagonalize a matrix.
We will review old exams on monday in ASB-213 near my office starting 2pm.
Updated 4/21/2014

Solutions and Comments on Quiz 16 from today
Today we talked about 5.2 and 5.3 (diagonalization, characteristic polynomial, etc.,).
Wednesday we will have a quiz for 20 points but it will be more like a test with a strict time limit of 10 minutes.
Please study 5.2 problems 11,15,19,21,22 and 5.3 problems 21-26 for the quiz.
Updated 4/21/2014

Solutions and Comments on Test 3 from today
Monday we will talk about 5.2 and 5.3 (diagonalization, characteristic polynomial, etc.,).
Monday we will have a quiz for 20 points but it will be more like a test with a strict time limit of 10 minutes.
Please study 3.3 problems 19-25 and 4.6 problems 19 to 25 for the quiz.
Updated 4/18/2014

TEST 3 (LAST BEFORE FINAL) ON FRI 18 APRIL
Review 12.10 to 1pm in Room 213 ASB-B Thu April 17 (NOT in usual classroom!).
Test will be on 3.1,3.2,3.3,4.1 to 4.7,5.1
Studying for test 3:
Please study class notes, review problems posted here, webwork problems.
Also try Spring 2011 Test III and Spring 2009 Test III (only problems 1,2,3 and 4 of test 3), and Spring 2009 Test II (all except problem 2).
Also try the quizzes 6,7,8 (only 1 and 2 of quiz 8) of spring 2011 at the SPRING 2011 AND SPRING 2009 update page
Please try the problems below before coming to review on Thursday.
3.1: 1,25,37
3.2: 1,3,21,25,27,29,31,33
3.3: 3,5,11,19,23
4.1: 1,7,8,11,13,21,23
4.2: 3,15,23,25
4.3: 1,11,15,21
4.4: 1,3,5,7,11,15.
4.5: 5,9,13,19,21.
4.6: 1,3,5,11,17.
4.7: 5,7,11,13.
5.1: 5,9,13,17,21,25,27.
Also try webwork sets "determinants" and "vector spaces."
Note that image of a transformation T is same as column space of its matrix A.
Kernel of T is same as Null Space of A.
Updated 4/11/2014.

Correction on a remark made in class:
The characteristic polynomial is det(A-λI) = (-1)n(λ-λ1)(λ-λ2)...(λ-λn)  (I was missing the power of -1).
You can see this by expanding along the last row or column and noting that the term (ann - λ) times its cofactor will contain the highest degree term λn times (-1)n.
Challenge (this is actually the challenge problem --problem 4 --from quiz 8 of spring 2011 at the SPRING 2011 AND SPRING 2009 update page ):
From det(A-λI) = (-1)n(λ-λ1)(λ-λ2)...(λ-λn) show that det(A) is the product of the eigenvalues counted with multiplicity.
This will be the same as the product of eigenvalues if they are all distinct.
Otherwise you include the multiplicity which is the number of times a root of the polynomial is repeated when you write the polynomial as a product of the λ - λi's.
Solutions and Comments on Quiz 15 from today.
Today we reviewed sections 4.6 and 5.1.
Tomorrow at 12pm in ASB-B 213 (near my office) we will review for test 3.
Please try the review problems posted above before coming.
Solutions and Comments on Quiz 14 from monday.
Updated 4/16/2014.

I have posted some problems on Webwork in the sets "determinants" and "vector-spaces."
You must not depend on webwork alone, though.
It doesn't have that many problems and the server can be annoyingly slow.
Updated 4/15/2014.

Due to power failure in the math dept could not post solutions.
Will post it wednesday but basically here is how you do it:
First you have to write the coordinate vectors of the basis polynomials in the standard basis {1,t,t^2}: for example 1-3t^2 is same as (1,0,-3).
Then you write the vector t^2 given by (0,0,1) as a linear combination of the three basis polynomials, say with coefficients a,b and c.
This will result in a matrix equation which you can solve using an augmented matrix and get the values of a,b and c.
The vector (a,b,c) is the coordinate vector of t^2 in the new basis.
Please study as many of the odd number problems from 5.1 as possible for wednesday's quiz.
Updated 4/14/2014.

Solutions and Comments on Quiz 13 from today.
Today we discussed sections 4.6 and 4.7.
Monday we will review them a bit and start to discuss 5.1.
4.6: 1,3,5,11,17,25.
4.7: 1, 5,7,11,13,19.
Updated 4/11/2014.

Today we discussed dimension, basis and coordinates (sections 4.4 and 4.5).
Please review problems below for friday's quiz:
4.4: 1,3,5,7,13,15,17,19,21,23,27,31,33.
4.5: 1,9,11,13,19,21,23,25,27,29,31.
Friday we will discuss 4.6 and 4.7.
Updated 4/9/2014.

Solutions and Comments on Quiz 12 from today.
Today we reviewed sections 4.1 to 4.3.
Wednesday we will discuss 4.4 and 4.5.
Updated 4/7/2014.

Today we discussed general vector spaces, their subspaces and basis vectors.
We looked at the example of the space of polynomials of degree less than or equal to n.
Monday we'll have a quiz based on 4.1 to 4.3.
Please see below for review problems from those sections.
Updated 4/4/2014.
Please try the following review problems from 4.1 to 4.3.
We will go over some of them in class tomorrow.
4.1: 1,3,5,7,13,21,23.
4.2: 3,7,17,25,31,35.
4.3: 3,11,19,21,29,33.
Updated 4/3/2014.

Today we reviewed chapter 3 and started chapter 4 -- vector spaces.
Updated 4/2/2014

Today we finished Chapter 3.
Wednesday we will start chapter 4, vector spaces.
We will have another quiz wednesday on the same topic but this time the matrix will not be given, only the determinant. You will have to use the properties of determinants.
Also try problems 23, 25, 27, 30,31 from 3.3.
Updated 3/31/2014.
For monday's quiz please try the following:
3.1: 1,3,9, 19,21,25,29,35 .
3.2: 3,5,11,19,21,25,27,29, 31,33,34,43.
3.3: 1,5,7,11,17.
Updated 3/28/2014.
Today we discussed further properties of determinants and how to find them using row reduction (3.1 and 3.2).
Friday we will finish chapter 3 as much as possible.
Solutions and Comments on Quiz 9 from today
Please try the following: 3.1: 1,3,9, 19,21,25,29,35 .
3.2: 3,5,11,19,21,25,27,29, 31,33,34,43.
Updated 3/26/2014.
Solutions and Comments on Test 2 from today
Monday we will start 3.1--determinanats.
Updated 3/21/2014

TEST II ON FRI, MAR 21
Review 12.10 to 1pm in Room 213 ASB-B Thu Mar 20 (NOT in usual classroom!).
Test will be on 1.8, 1.9, 2.1,2.2,2.3,2.8,2.9
Studying for test 2:
Please study class notes, review problems posted here, webwork problems.
Also try Spring 2011 Test II , Spring 2011 Test III (only problems 1 and 2 from test II) and Spring 2009 Test II
Also try the quizzes 3,4,5,6 of spring 2011 at the SPRING 2011 AND SPRING 2009 update page
Ignore any problem involving determinants or Cramer's rule.
Please try the problems below before coming to review on Thursday.
1.8: 3,21,31,35
1.9: 1,17,23,37
2.1: 1,5
2.2: 7,9,21,31
2.3: 1,5,11,35
2.8: 5,11,21,23
2.9: 1,9,11,13,17,19,23.
Also try webwork sets "linear-transformations" and "matrices."
Note that image of a transformation T is same as column space of its matrix A.
Kernel of T is same as Null Space of A.

Please note that tomorrow we will do a review for the test on friday.
it will be at noon in ASB-B 213 (NOT our usual classroom).
Solutions and Comments on Quiz 8 from today
Updated 3/19/2014.
Hope you are enjoying the extended spring break.
We will have the test as scheduled on friday, but only upto 2.9.
3.1 and 3.2 are not included, so you don't need to study determinants and Cramer's rule.
Everything else remains the same.
I will be in my office tomorrow (tuesday) from 2 to 5pm if anyone needs help.
Updated 3/17/2014.
I have added problems to webwork set "linear transformations."
Solutions and Comments on Quiz 7 from yesterday
Updated 3/7/2014.
MIDTERM GRADES HAVE BEEN POSTED ON BISONWEB.
If you got a U (unsatisfactory) it means you are in danger of getting a grade below C in the class and you need to do much better in future tests and quizzes.
Today we had a special review session. The aim was to give people practice with the more conceptual and abstract type of problems.
Here are some problems of this type for practice:
1.8: 25,27,31
1.9: 23,24
2.2: 8,9,10,11,16,24
2.3: 11,12,20,23,35
2.8: 21,22,33,35
2.9: 17,18,25.
I also gave Quiz 7 but will post the solution tomorrow after everyone has had a chance to try it.
Updated 3/6/2014
Today we discussed how to find inverse matrices using row reduction.
Tomorrow (thursday) we will have a review session from 12.10 to 1pm in room 213 of ASB-B.
We will also have a quiz (Quiz 7) tomorrow on 2.2 and 2.3.
2.2: 1, 5,9,11,12,13,21, 31.
2.3: 1,3,5,11,13,17,18, 21,33,35.
Updated 3/5/2014.

Solutions and Comments on Quiz 6 from today
Monday we will talk about inverse matrices and also start 2.8.
Please try the review problems listed below (2/26 update) from 1.9,2.1,2.2 and 2.3.
Updated 2/28/2014

Today we continued to talk about linear transformations 1.9.
We also talked about matrix operations, including inverse and transpose.
Friday we will finish 2.1 to 2.3 and have quiz 6.
1.9: 1,3,5,8,17,19,21,23,35
2.1: 1,9,23,25.
2.2: 11,13,21.
2.3: 11,18.
Please try webwork set "linear transformations."
Updated 2/26/2014
Solutions and Comments on Quiz 5 from today
Updated 2/24/2014

Today we continued to talk about linear transformations (1.8 & 1.9).
Monday we will have finish 1.8 and 1.9 and have quiz 5.
Please try 3, 13, 21, 17,19, 23,27, 33,35 from 1.8.
Please also try 17,19, from 1.9.
Please try webwork set "linear transformations."
Updated 2/21/2014
Today we talked about linear transformations (1.8).
Friday we will have continue with 1.8 (no quiz).
Please try 17,19, 23,33,35 from 1.8.
Updated 2/19/2014
Solutions and Comments on Test 1 from today
Updated 2/12/2014

TEST 1 ON WED, FEB 12
(Notice the corrected date; it is not thursday)
Review on monday, feb 10
Studying for test 1:
Please study class notes, review problems posted here, webwork problems.
Also try Spring 2011 Test I and Spring 2009 Test I
(in 2009 test I all except 6 and 8 (the challenge problem))
Also try the quizzes before test 1 in both spring 2011 and 2009 at the SPRING 2011 AND SPRING 2009 update page
Please try the problems below before coming to class on monday.
1.1: 7,11,32
1.2: 3,11,21
1.3: 11,15,23
1.4: 7,11,21,23
1.5: 7,13,23
1.6: 3,5,9
1.7: 1,5,21

Solutions and Comments on Quiz today
Quiz was cancelled because problem was not stated correctly, although it could still have been solved as stated.
Today we finished talking about linear independence (section 1.7).
Monday we will do a review for the test.
Please do the following review problems posted at the top
Updated 2/7/2014

Today we talked about sections 1.6 and 1.7.
Please do the following review problems:
1.6: 3,5,7,11,13,15
Try the application problems in the webwork set "matrix-equations."
Updated 2/5/2014

Solutions and Comments on Quiz 4 today

Today we finished talking about sections 1.3,1.4 and 1.5.
Wednesday we will move on to 1.6
Please do the following review problems:
1.3: 1,5,11,13,23
1.4: 1,3,11,15,17,19,23
1.5: 1,7,13,25,27,33,35,37,38,39
I have added problems to the webwork set "matrix-equations."
Updated 2/3/2014

Solutions and Comments on Quiz 3 today

Today we finished talking about sections 1.3,1.4 and 1.5, specifically matrix equations and vector equations.
Monday we will review these sections a bit and move on to 1.6
Please do the following review problems:
1.3: 1,5,11,13,23
1.4: 1,3,11,15,17,19,23
1.5: 1,7,13,25,27,33,35,37,38,39
I have added problems to the webwork set "matrix-equations."
Updated 1/31/2014

Today continued about the basics of using matrix or vector equations to represent systems of equations.
Please try problems 25,27,33,35,37,38,39 in section 1.5.
Please also try problems 1,3,11,15,33,35 in section 1.4.
On friday we will continue with 1.3,1.4 and 1.5.
Updated 1/29
Today we went over the basics of using matrix or vector equations to represent systems of equations.
I presented the similarities between solutions for one equation in two variables, (whose graph is a line), with those of more general systems.
Please try problems 1,3,11,15,33,35 in section 1.4.
On wednesday we will continue with 1.3 and 1.4.
Updated 1/27
Solutions and Comments on Quiz 2 today

Today we finished talking about section 1.2, specifically echelon and reduced echelon matrices and pivots and how to tell what kind of solutions a system has.
Updated 1/24/2014

Today we talked about section 1.2, specifically echelon and reduced echelon matrices and pivots.
Please do these review problems in 1.2 for friday's quiz:
3, 7,11,15,21,23
Please try the problems in webwork set "1.2-echelon-matrices"
I could only find one problem on webwork on this topic, so please also do the book problems.
Updated 1/22/2014

Today we continued discussing 1.1.
We solved a 3 by 3 system that was inconsistent using augmented matrix.
Solutions and Comments on Quiz 1 today

Please do these review problems in 1.1: 5,7,9,13,17,23,27.
I have added more problems to the webwork set "1.1-solving-systems."
Updated 1/17/2014

Please also try the webwork set "1.1-solving-systems" for friday's quiz.
Updated 1/16

Today we started with chapter 1, section 1
We discussed the elementary row operations and how they can be used to solve a system of equations
These simple operations have the advantage that they can be used to systematically solve any system no matter how many equations or variables are involved.
Please try problems 1,2,3,4,17 and 24
We will have a very simple quiz on wednesday based on these.
Updated 1/15/2014

CLASSROOM SWITCH:
In order to accomodate an older Prof who has difficulty with the stairs, we will be moving to room 306 in same building (Chem bldg) starting wednesday.