Information for Math 330-1: Linear Algebra
Spring 2012

Likely grading scale: 640 guarantees an A, 530 guarantees a B, and 420 guarantees a C. I will look at what happens if your worst test score is replaced by half your score on the final, though I won't guarantee that this will definitely be an alternative method of calculating your grade.

I will have office hours Sunday 2-4 (or until the last person leaves or I get too hungry) in the library, 1st floor towards the back.

Review material that has been posted through the semester.

As announced in class, I have decided to have you turn in reading guides and homework instead of having quizzes.

The information on the syllabus appears below. It is also available as a single PDF file. Jump to the Reading Guides or the Homework.
Lectures: MWF 12:30-1:20am, TLC 222
Instructor: Alexander Woo, Brink Hall 312, phone: 885-6741, awoo@uidaho.edu
Office hours: Monday 9-10, Tuesday 3-4, Wednesday 2:30-4, Thursday 1:30-3, and by appointment. These are the times I promise to be in my office and available. You are also welcome at any time to check if I am available, either in person or by telephone.
Objecives:
  • To improve the ability to reason abstractly and quantitatively and the ability to communicate such reasoning and its results.
  • To understand precisely the vocabulary of linear algebra (and there is a lot!) from both algebraic and geometric viewpoints.
  • To understand some ways linear algebra can be used, both outside and within mathematics.
Text: David C. Lay, Linear algebra and its applications, 4th edition.
Grading: Points are given as follows:
Homework completion 50 points
Quizzes 10 points per quiz
Tests 100 points per test (there are 3)
Final 200 points
There will be about 35 quizzes, so a total of 900 points. You should expect that about 780 points will be needed for an A, 660 points for a B, and 540 points for a C. The points for an A is based on an expectation of about 90 percent for the quizzes and homework and 85 percent for the tests and final.
Homework: There are two separate components to the homework.
First, you are expected to read the section we will cover before class. To help guide your reading, I will give you several reading guide questions which are conceptual in nature which you should try to answer for yourself while doing the reading.
Second, for each class, I will assign problems based on the section we covered that class. You are expected to attempt all of the problems before the next class.
Your homework will be evaluated in two ways. First of all, each class will have a quiz with questions from the homework due the previous class (and assigned two classes ago). Second, with each test you will turn in your homework notebook (or two if you prefer to keep reading guides and problems separately) which will be evaluated (for completeness only) and returned to you the next class.
Quizzes: Each quiz will include one reading guide question and one homework problem, both assigned two classes ago and possibly discussed (if you asked about it) the previous class. (This means the reading guide question will be about the material discussed the previous class, and the homework problem will be about the material discussed two classes ago.) The quizzes are open notes but not open book. Notes includes anything that you write or type yourself prior to the beginning of class. I will try to return quizzes the next class but will get behind at certain points during the semester.
Exams: There will be three tests in class, tentatively on February 8, March 9, and April 16. You should let me know about any conflicts preventing you from taking a test in class on the scheduled dates at least one week in advance. Make-up tests will only be given for documented, important conflicts in accordance with the one week policy, or for genuine documented emergencies.
The final exam will be on Monday, May 7 at 12:30. Requests to take the final at a different time must be made in writing and be approved by me, the department chair, and the dean. Except in the case of a documented emergency, missing the final exam will result in a grade of F.
Student Disabilities: Reasonable accomodations are available for students with documented temporary or permanent disabilities. All accomodations must be approved through Disability Support Services in order to notify your instructor(s) as soon as possible regarding accomodation(s) needed for the course.
If you have a disability of some kind which requires accomodation, please talk to Disability Support Services as soon as possible. If there is anything I can do to help you, please talk to me as well.

Reading guides

Dates listed are the dates we discuss the material covered by the reading guide. The quiz drawing from the reading guide takes place the class following.
Linear Equations (1.1) (January 13)
  1. What is an system of linear equations? Give an example of a system of linear equations. Give an example of a system of equations in 3 variables that is not linear.
  2. What are the coefficient matrix and augmented matrix for a system of linear equations?
  3. Describe briefly in general terms how to solve a system of linear equations.
  4. What are elementary row operations?
  5. What do existence and uniqueness mean in the context of solutions to linear systems?
Echelon Form (1.2) (January 18)
  1. Give examples of matrices with 3 rows and 6 columns that are:
    • In reduced echelon form.
    • In echelon form but not reduced echelon form.
    • Not in echelon form.
  2. What are the pivot columns of a matrix in echelon form?
  3. What does it mean for two matrices to be row equivalent?
  4. Describe briefly how to take a matrix A and produce a matrix which is in echelon form and row equivalent to A.
  5. What are basic variables? What are free variables? How do you tell which are which from a matrix in (reduced) row echelon form?
  6. What features of the row echelon form (for the augmented matrix) tell you if a system of linear equations has a solution?
  7. What features of the row echelon form (for the augmented matrix) tell you if a system of linear equations has only one solution?
  8. What features of the row echelon form for the coefficient matrix will guarantee there will be a solution no matter what is on the right hand side of the equations?
Vector Equations (1.3) (January 20)
  1. What are vectors algebraically?
  2. How do we interpret vectors geometrically?
  3. How do we interpret addition and scalar multiplication of vectors geometrically?
  4. What is a linear combination of vectors?
  5. Given some vectors v1, ..., vp, what is the span of the vectors?
  6. How can one geometrically describe the span of 1 nonzero vector in R3? What about the span of two nonzero vectors in R3, neither of which is a multiple of the other?
Ax=b (1.4) (January 23)
  1. We now have three ways of writing a system of linear equations.
    • What are they?
    • Write a system of 3 equations in 5 variables in each of the 3 ways.
  2. How can one rephrase the question of whether Ax=b is consistent in terms of the concept of span?
  3. Suppose we have a matrix A such that Ax=b is always consistent, no matter what b is (as long as b has the right number of entries to make sense). What does this mean about the span of the columns of A? How is this related to the (reduced) row echelon form for A?
  4. Given numbers in A and x, what is another way to compute Ax which does not add vectors?
  5. What distributive law applies to multiplication of a matrix and a vector?
Solution sets (1.5) (January 25)
  1. What makes a system of linear equations Ax=b homogeneous?
  2. Relate the solutions of a homogeneous equation to the notion of span.
  3. What does the number of free variables mean as far as span is concerned (for a homogeneous system)?
  4. What is the translation of a plane by a vector?
  5. How is the set of solutions to a nonhomogeneous equation Ax=b related to the solutions to the homogeneous equation Ax=0 (with the same A)?
Applications of linear systems(1.6) (January 27) Read the entire section - no reading guide questions
Linear independence (1.7) (February 1)
  1. What does it mean for a set of vectors to be linearly independent? To be linearly dependent? Given a set of vectors, what equation do you study the solutions of to tell if they are linearly independent or dependent?
  2. How can one easily tell if a set of just one vector is linearly independent? What about a set of just two vectors?
  3. Given an example of 3 vectors in R3 that are linearly dependent, but none of which is a multiple of another.
  4. A set of vectors is automatically linearly dependent if it has too many vectors. How many is too many?
  5. A set of vectors is automatically linearly dependent if one of the vectors equals a particular vector. Which particular vector?
Linear transformations (1.8) (February 1)
  1. What is a function from Rn to Rm?
  2. What is a matrix transformation?
  3. What makes a function from Rn to Rm a linear transformation?
  4. Describe the possible images of a line under a linear transformation.
Matrix of a linear transformation (1.9) (February 3)
  1. Suppose I have a linear transformation defined geometrically or by a bunch of arrows (as in class today). How do I find the matrix for that linear transformation?
  2. What does it mean for a function to be onto? One-to-one?
  3. How are being onto or one-to-one related to the span of the columns or linear independence of the columns?
Matrix operations (2.1) (February 10)
  1. How does one add matrices?
  2. What does multiplication of matrices correspond to in terms of mappings or linear transformations?
  3. How is the product AB defined in terms of the columns of B?
  4. How can you think of the entries of AB in terms of obtaining it from a row of A and a column of B?
  5. What does it mean when it is said that matrix multiplication does not commute?
  6. What does it mean when it is said that cancellation does not hold for matrix multiplication?
  7. What is the transpose of a matrix?
  8. How does transpose interact with multiplication?
Matrix inverses (2.2) (February 13)
  1. What does it mean for a matrix to be invertible?
  2. What does it mean for a matrix C to be the inverse of some matrix A?
  3. Assuming A and B are invertible, is AB invertible? If so what is its inverse?
  4. Given an actual matrix, how does one actually find its inverse (or find that it has none)?
Invertible matrices (2.3) (February 15)
  1. Learn all of Theorem 8. Remember it.
  2. What is the definition of a linear transformation being invertible? How is it related to a matrix being invertible?
Block matrices (2.4) (February 17)
  1. If we have two matrices which are divided into several parts, how can we think about the product in terms of these parts?
  2. Describe the column-row expansion of a product
  3. What is a block diagonal matrix? How does its inverse compare to the inverses of the blocks?
Computer graphics (February 24)
  1. What are homogeneous coordinates?
  2. What advantage does homogeneous coordinates provide?
  3. What is a perspective projection?
Subspaces (February 29)
  1. What three properties does a subspace have to satisfy?
  2. Give two examples each of subsets of R2 and R3 that are subspaces.
  3. Give two examples each of subsets of R2 and R3 that are not subspaces.
  4. What is the column space of a matrix?
  5. What is the null space of a matrix?
Basis and dimension (March 2)
  1. If I give you a subspace H and a bunch of vectors v1, ... , vp, how do you test if these vectors form a basis for H?
  2. Given a matrix A, how do you find a basis for Nul A?
  3. Given a matrix A, how do you find a basis for Col A?
  4. (From Section 2.9) What is the dimension of a subspace? Given a matrix A, how do you find the dimensions of Nul A and Col A?
Coordinates and rank (March 5)
  1. What is the rank of a matrix?
  2. What do the rank and the dimension of the null space add up to?
  3. What does the basis theorem tell you about sets of vectors of the right size?
  4. What are the coordinates for a vector relative to a basis?
Markov chains (4.9) (March 19)
  1. What is a Markov chain?
  2. What is a state vector?
  3. Suppose the entries of the state vector are the populations of the 50 states. What do the entries of the stochastic matrix mean?
  4. What is the steady-state vector for a Markov chain?
  5. How is the long term behavior of the Markov chain related to the steady-state vector?
Determinants (3.1) (March 26)
  1. What is (i,j)-cofactor of a matrix?
  2. How does one use the cofactors of a matrix to calculate a determinant?
  3. Does it matter which row or column is used?
  4. What is the determinant of a triangular matrix?
Properties of determinants (3.2) (March 28)
  1. How does doing the three basic row operations change the determinant of a matrix?
  2. How can one use row operations to more easily calculate a determinant?
  3. What does the determinant tell us about the invertibility of a matrix?
  4. What is the determinant of a product of matrices in terms of the determinants of the original matrices?
Eigenvalues and eigenvectors (5.1) (March 30)
  1. Given a matrix A, what is an eigenvalue of A? What is an eigenvector of A?
  2. What vector is not allowed by definition to be an eigenvector?
  3. What is the eigenspace of A corresponding to an eigenvalue?
  4. What does multiplying by A do geometrically to points in an eigenspace of A?
  5. Why is it easy to find eigenvalues for a triangular matrix?
  6. What do we know about linear independence for eigenvectors corresponding to different eigenvalues?
Finding eigenvalues and eigenvectors (5.2) (April 2)
  1. What is the characteristic polynomial of a square matrix?
  2. How can one use the characteristic polynomial to find eigenvalues?
  3. Given an eigenvalue, how does one find the eigenvectors associated to that eigenvector?
  4. What is the algebraic multiplicity of an eigenvalue?
  5. What does it mean for two matrices to be similar?
  6. What do we know about the characteristic polynomials of similar matrices?
Diagonalization (5.3) and Coordinates (4.4) (April 6)
  1. How does diagonalization help in finding large powers of a matrix?
  2. What does it mean for a matrix to be diagonalizable?
  3. How does one find a diagonalization from the eigenvalues and eigenvectors of a matrix?
  4. What about the eigenvalues guarantees that a matrix will be diagonalizable?
  5. If the above condition does not hold, is there a condition on the factors of the characteristic polynomial and the dimensions of the eigenspaces that guarantees diagonalizability?
  6. What is a change of coordinates matrix?
  7. Given a basis, how does one get the change of coordinates matrix from it to the standard basis?
  8. Given a basis, how does one get the change of coordinates matrix from the standard basis to it?
Abstract vector spaces (4.1 and 4.2) (April 11)
  1. Without being precise and listing the 10 axioms: what is a vector space?
  2. Give 3 examples of vector spaces.
  3. What is a subspace of a vector space?
  4. For each of your examples of vector spaces, give one subspace of that vector space.
  5. What is the span of a set of elements of a vector space?
  6. What is a spanning set for a subspace?
  7. What is a linear transformation?
  8. What is the kernel of a linear transformation?
  9. What is the range of a linear transformation?
  10. Give an example of a linear transformation between vector spaces which are not Rn, and give the kernel and range.
Basis and dimension (4.3 and 4.5) (April 18)
  1. What does it mean for a set of objects in a vector space to be linearly independent?
  2. What does it mean for a set of objects in a vector space to be a basis of the vector space?
  3. Give an example of a vector space other than Rn and a basis for it.
  4. What is the dimension of a vector space?
  5. What does it mean for a vector space to be infinite dimensional?
  6. Give an example of an infinite dimensional vector space.
  7. What do we know about the dimensions of a vector space and any of its subspaces?
  8. What does the basis theorem tell us about sets of d objects in a d-dimensional vector space?

Homework

Dates listed are the dates I answer questions about the homework, presumably after you have attempted the problems. The quiz drawing from the homework takes place the class following.
Linear Equations (1.1) (January 18) Section 1.1: 1, 2, 7, 8, 13-16, 27-30
Echelon Forms (1.2) (January 20) Section 1.2: 3-6, 9, 12, 15, 16, 23, 26
Vector Equations (1.3) (January 23) Section 1.3: 6-10, 12, 17, 18, 24, 28ab, 32. Explain what the span of the two vectors mean in problem 28 (assuming the plant can somehow burn negative amounts of coal).
Ax=b (1.4) (January 25) Section 1.4: 3, 4, 9, 10, 13, 17, 19, 30-33, 35
Solution sets (1.5) (January 27) Section 1.5: 1, 2, 9-11, 15, 16, 27-31
Applications of linear equations (1.6) (January 30) Section 1.6: 1, 2, 5, 6, 7, 12, 15
Linear independence (1.7) (February 1) Section 1.7: 1, 2, 6, 16-18, 33, 38
Linear transformations (1.8) (February 3) Section 1.8: 1-4, 9, 17, 18, 27, 36
Matrices for linear transformations (1.9) (February 6) Section 1.9: 3, 4, 9, 10, 13, 17, 18, 26, 29, 35
Matrix operations (2.1) (February 15) Section 2.1: 1, 2, 5-8, 14, 18, 19, 24
Matrix inverses (2.2) (February 15) Section 2.2: 1, 2, 10, 21, 22, 31, 32
Invertible matrices (2.3) (February 17) Section 2.3: 1, 2, 7, 8, 12-15, 17, 18, 36
Block matrices (2.4) (February 22) Section 2.4: 1, 2, 7, 8, 13, 15, 21
Computer Graphics (2.7) (February 29) Section 2.7: 3-6, 15-20
Subspaces (2.8) (March 2) Section 2.8: 1, 2, 7-12, 31, 32
Basis and Dimension (March 5) Section 2.8: 15, 16, 19, 20, 23, 24, 27, 28 Section 2.9: 9, 10
Coordinates and Rank (March 7) Section 2.9: 3-6, 19, 20, 27, 28
Markov Chains (4.9) (March 21) Section 4.9: 1, 2, 6, 7, 11, 12
Determinants (3.1) (draft due March 28; turn in March 30) Section 3.1: 1-4, 9, 10, 13
Properties of determinants (3.2) (draft due March 30; turn in April 2) Section 3.2: 1, 2, 6, 7, 15, 16, 24, 29, 31
Eigenvalues and eigenvectors (5.1) (draft due April 2; turn in April 4) Section 5.1: 4, 5, 7, 8, 15, 16, 23, 31, 35
Finding eigenvalues and eigenvectors (5.2) (draft due April 4; turn in April 6) Section 5.2: 2-4, 11, 12, 19, 27
Diagonalization and coordinates (5.3 and 4.4) (draft due April 9; turn in April 11) Section 5.3: 5, 6, 9, 10, 14, 15, 31 Section 4.4: 2, 3, 6, 7, 9, 10
Abstract vector spaces (4.1 and 4.2) (draft due April 16; turn in April 18) Section 4.1: 5, 6, 8, 21, 22
Section 4.2: 3, 6, 30, 31, 32
Worksheet: 1, 5, 6, 7, 9, 11, 12, 15
Basis and dimension (4.3 and 4.5) (draft due April 20; turn in April 23) Section 4.3: 3, 4, 23-26, 31, 33, 34
Section 4.5: 9, 10, 21, 25, 26