Spring 2010, Math 204, Section 01.

Linear Algebra

Course Information

Lowest score

I will drop the lowest of your test and homework scores and replace it with the next lowest scores. So if h, e1, e2 denote these scores you will get max(h,e1,e2)+2 * second largest of (h, e1, e2). The final exam score will not be dropped.

Lecture schedule

This schedule is dynamic and is updated nearly every weekday

The section(s) that correspond to the lecture topic is(are) indicated in parentheses.

I mentioned in class that you should submit all the problems. However, I have updated this policy. You should submit the problems listed in the homework section.

The comments column will include deadlines for homework and reminders of exams and reviews.

Date Topic Problems Comments
Wednesday, 1/13/2010 Introduction to linear systems (1.1) 1.1: 4, 7, 11, 18, 25, 28, 31, 35, 40.
Friday, 1/15/2010 Linear systems and Gaussian elimination (1.2) 1.2: 6, 17, 18, 19, 23, 27, 32, 34, 38.
Monday, 1/18/2010 Matrix algebra and qualitative properties of linear systems (1.3). 1.3: 2 – 6 (even), 14 – 20 (even), 24, 30, 34, 35, 38.
Wednesday, 1/20/2010 Matrix vector multiplication, dot products and linear combinations (1.3)
Friday, 1/22/2010 Linear transformations (2.1)
  • 2.1: 2, 4, 6, 12, 13, 20, 34, 42
  • 2.1 optional: 52 — 54
Homework 1 is due.
Monday, 1/25/2010 Geometry of linear transformations (2.1, 2.2) 2.2: 2, 6, 10, 12, 22, 32, 38, 47.
Wednesday, 1/27/2010 Projections (2.2) Read the examples of reflections and rotations in section 2.2.
Friday, 1/29/2010 More examples of linear transformations.
  • Homework 2 is due.
  • Homework 1 solutions posted below
Monday, 2/1/2010 Matrix multiplication and composition of linear transformations (2.3). 2.3: 1 — 12 (even), 16, 18, 21, 22, 26, 29, 31, 32, 46, 56.
Wednesday, 2/3/2010 The inverse of a linear transformation (2.4). 2.4: 2, 13, 14,19, 28, 29, 33, 34, 35, 37, 42, 43, 45, 46, 48. We're beginning this section. We will finish on Friday. Homework 3 is now due next Friday, 2/12/10.
Friday, 2/5/2010 Inverses of linear transformations (2.4).
Monday, 2/8/2010 Induction. Introduction to subspaces (3.1). Here are some notes on induction.
Wednesday, 2/10/2010 Examples of matrices. We're going to look at some examples of matrices, and invertibility. Here is a pdf file containing some examples. I will update and improve it as we go along.
Friday, 2/12/2010 Subspaces, range and null space of a linear transformation (3.1) 3.1: 5, 9, 15, 19, 24, 31, 35, 36, 39. Homework 3 is due.
Monday, 2/15/2010 How to compute the dimension and a basis for the null space (part of 3.2). Try to work out some more examples of your own.
Wednesday, 2/17/2010 How to compute the dimension and a basis for the range. (part of 3.3) 3.2: 1, 2, 5, 6, 27 — 33; 3.3: 3, 9, 17, 23, 24, We have skipped over the results on linear independence, dependence and dimension. We will return to these topics after exam 1.
Friday, 2/19/2010 Review for exam 1. Bring questions, misconceptions, and doubts.
Monday, 2/22/2010 EXAM 1 Homework 4 is due.
Wednesday, 2/24/2010 Linear dependence and independence (3.2, 3.3)
  • 3.2: 11, 13, 15, 17, 19, 21, 24, 39, 42, 49, 51.
  • 3.3: 27, 28, 29, 32, 33, 34, 60, 64.
 We will define the notion of dependence, spanning set, and linearly independent set. We'll prove some theorems as well.
Friday, 2/26/2010 Bases and coordinates (3.3, 3.4) Trying to prove that the dimension makes sense.
Monday, 3/1/2010 Bases and coordinates (3.3, 3.4) Finishing the proof of the uniqueness of dimension and moving on to coordinates. Here is a typed up version of the proof from today's class.
Wednesday, 3/3/2010 Bases and coordinates (3.4) 3.4: 3, 9, 13, 18, 21, 27, 31 — 36, 43, 47, 48, 58, 70, 71.
Friday, 3/5/2010 Similar matrices (3.4)
Monday, 3/15/2010 Orthonormal Bases (5.1) 5.1: 7, 12, 13, 17, 22, 32 — 36, 40, 41.
Wednesday, 3/17/2010 Projections and the Gram-Schmidt process (5.2) 5.2: 3, 4, 9, 31 — 34.
Friday, 3/19/2010 Orthogonal Transformations (5.3) 5.3: 1 — 11, 33, 34, 35, 37, 43.
Monday, 3/22/2010 Symmetric matrices (5.3). 5.3: 13 &mdash 26; 27, 47. Read the part of chapter 5.3 on page 214, 215 concerning transposes and the inverse of an orthogonal matrix.
Wednesday, 3/24/2010 Least squares (5.4). 5.4: 1 &mdash 7; 16, 19 — 25.
Friday, 3/26/2010 More least squares
Monday, 3/29/2010 Linear spaces (4.1). 4.1: 1 — 45 (odd). These are the problems for today and 3/31/2010. Homework 8 will be due on Friday, 4/9/2010.
Wednesday, 3/31/2010 Vector (linear) spaces (4.1)
Friday, 4/2/2010 Review for exam 2. The fact sheet has been updated for exam 2.
Monday, 4/5/2010 EXAM 2
Wednesday, 4/7/2010 Linear transformations on linear spaces (4.2). 4.2: 1, 4, 9, 14, 23, 28, 32, 35, 54, 54, 66, 68, 69, 70, 78,
Friday, 4/9/2010 Linear transformations (4.2) 4.2: See above. Solutions to exam 2 are posted below.
Monday, 4/12/2010 Eigenvectors and eignevalues, an intro
Wednesday, 4/14/2010 Determinants (6.1, 6.2) 6.1: 1, 6, 10, 13, 21, 22, 23, 29, 43, 44.
Friday, 4/16/2010 6.2: 2, 3, 11, 13, 14, 15, 29, 32, 35. Professor Willett will work through some more of the theory of determinants in relation to eigenvector and eigenvalue calculations. My Friday office hours are cancelled. I will have office hours on Monday 4/19/2010 (11 - noon).
Monday, 4/19/2010 Eigenvalues and eigenspaces (7.2) 7.2: 1, 3, 6, 8, 14, 15, 19, 21, 30, 34.
Wednesday, 4/21/2010 Eigenvectors (7.3) 7.3: 1 — 13 (odd), 19, 20, 22, 27, 28, 31, 32 — 34.
Friday, 4/23/2010 Diagonalization (7.4) 7.4: 1, 5, 11, 17, 25, 26, 32, 35, 37, 38.
Monday, 4/26/2010 Symmetric matrices and the spectral theorem (8.1). 8.1: 7, 10, 12, 14, 16, 23.
Friday, 4/30/2010 FINAL EXAM

Exams

  • Exam 1
    • Monday, February 22nd, 2010, 1:10 pm — 2:00 pm
    • Content: Chapters 1, 2, 3.1 — 3.3. (linear independence and dependence will not be tested, but you should be able to compute a basis for the range and null-space)
    • Review: Friday, 2/22/2010, in-class. Here is a little fact sheet
    • Practice exam: PDF.
  • Exam 2
    • Monday, April 5th, 2010, 1:10 pm — 2:00 pm
    • Content: 3.3 — 4.1, 5.1 — 5.4.
    • Review: Friday, April 2nd, 2010 in class.
    • Practice exam: PDF.
    • Solutions to actual exam: PDF
  • Final Exam
    • Friday, April 30th, 2010, 9:00 am — 11 am in SC 2212 (our classroom). Official schedule
    • Content: Chapters 1, 2, 3, 4.1 — 4.2, 5.1 — 5.4, 6.1 — 6.2, 7.2 — 7.4, 8.1.
    • Review: Wednesday, April 28th, 2 pm — 4pm in SC 2212.
    • Practice exam: PDF. Solutions: PDF

Homework

Additional material

  • Induction. We discussed proof by induction in class. Here are some short notes on induction together with a few exercises for you to try.
  • Examples of matrices. We discussed different examples of matrices in class. I was tempted to call them patterns. Here are some short notes on the different examples. There are some exercises as well which you should try if you have time, since this will give you a better feeling for matrices. I will update the notes as the semester progress.
  • Dimension of a subspace. Here is a typed-up version of the proof of central result that is required to show that dimension of a subspace is well-defined.