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Reading Assignments - Math 221 Linear Algebra - Fall 2015

Be sure to check back, because this may change during the semester.

All numbers indicate sections from Linear Algebra and Its Applications, 4th Edition by Lay

For Thursday September 3

Section 1.1 Systems of Linear Equations
Section 1.2 Row Reduction and Echelon Forms
Section 1.3 Vector Equations

To read: All

Reading Questions

Let u=(1,2,-1) and v=(-3,1,5).

  1. Is A in reduced echelon form? Why or why not?
  2. What are the pivot columns of A? What are the values of the pivots of A?
  3. If B is the augmented matrix of a linear system, does the system have any free variables? If so, what are they?
  4. Write the vector w=(-3, 8, 7) as a linear combination of u and v.

Submit answers through onCourse

For Tuesday September 8

Section 1.4 The Matrix Equation Ax=b
Section 1.5 Solution Sets of Linear Systems

To read: All

Reading Questions

  1. Let
    a. Does Ax = b have a solution for this specific value of b? Why or why not?
    b. Does Ax = b have a solution for every b ∈ ℝ3? Why or why not?
  2. Explain the difference between a homogeneous system of equations and a non-homogeneous system of equations.
  3. If the system Ax = b is consistent and Ax = 0 has a non-trivial solution, how many solutions does Ax =b have?

Submit answers through onCourse

For Thursday September 10

Section 1.7 Linear Independence

To read: All

Reading Questions Let

  1. Is the set of vectors {u, v} linearly independent or linearly dependent? Explain.
  2. Is the set of vectors {u, v, w} linearly independent or linearly dependent? Explain.
  3. If the vectors u, v, w, and z form the columns of the matrix A, does Ax = 0 have a non-trivial solution? Explain.

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For Tuesday September 15

Section 1.8 Introduction to Linear Transformations

To read: All

Reading Questions

  1. Suppose T: ℝ4 → ℝ5 is a linear transformation and T(x) = Ax for some matrix A. What are the dimensions of A? Why?
  2. Give an example of a function T: ℝ2 → ℝ2 that is not a linear transformation. Explain.

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For Thursday September 17

Section 1.9 The Matrix of a Linear Transformation

To read: All

Reading Questions

  1. Give the matrix A for the linear transformation T:ℝ2 → ℝ2 that expands vertically by a factor of 4.
  2. Let T:ℝ5 → ℝ3 be a linear transformation with standard matrix A where A has three pivots. Is T one-one? Explain.
  3. Let T:ℝ3 → ℝ5 be a linear transformation with standard matrix A where A has three pivots. Is T one-one? Explain.

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For Tuesday September 22

Section 2.1 Matrix Operations
Section 2.2 Inverse of a Matrix

To read: All

Reading Questions

  1. Give one way in which matrix multiplication differs from multiplication of real numbers.
  2. Let
    Compute the following products or explain why it is impossible: AB, BA, BAT
  3. Let . Find B-1.
  4. Give an example of a 2 x 2 singular matrix.
  5. If A is invertible, how many solutions does Ax=b have? Why?

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For Thursday September 24

Section 2.3 Characterizations of Invertible Matrices

To read: All

Reading Questions

  1. If A is an n x n matrix with n pivots, how many solutions does Ax=b have? Why?
  2. If A is an invertible matrix, are the rows of A a linearly independent set or a linearly dependent set? Why?

You should think about these, but no need to send in your answers through onCourse.

For Tuesday September 29

Exam 1 tonight No Reading Assignment.

For Thursday October 1

Section 3.1 Introduction to Determinants
Section 3.2 Properties of Determinants

To read: All

Reading Questions

  1. Let . Find det(A).

  2. Let . Find det(B).
  3. Why do we care about finding the determinant of a matrix?

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For Tuesday October 6

Section 4.1 Vector Spaces and Subspaces
Section 4.2 Null Spaces, Column Spaces, and Linear Transformations

To read: All

Reading Questions

  1. Give an example of a subset of ℝ2 that is not a subspace of ℝ2.
  2. Let ℙ4 denote the set of all polynomials of degree 4 or less with real coefficients. Give examples of two vectors in the vector space ℙ4.
  3. If the columns of A are linearly independent, what is Nul(A)? Why?
  4. If A is 6 x 9 with 6 pivots, what is Col(A)? Why?

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For Thursday October 8

No class today. I'll be at a conference.

For Tuesday October 13

Fall Break. No Reading Assignment.

For Thursday October 15

Section 4.3 Linearly Independent Sets; Bases

To read: All

Reading Questions

Let

  1. Do the columns of A form a basis for ℝ3? Explain.
  2. Give a basis for Col(A).
  3. Give a basis for Nul(A).

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For Tuesday October 20

Re-read the section, but no Reading Questions for today.

For Thursday October 22

Section 4.4 Coordinate Systems
Section 4.5 The Dimension of a Vector Space

To read: All

Reading Questions Let

  1. Show that B is a basis for ℝ2.
  2. Find the coordinate vector [x]B of x relative to B.
  3. What is the dimension of ℝ3? Why? Does this make sense geometrically?

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For Tuesday October 27

Exam 2 tonight. No Reading Assignment.

For Thursday October 29

Section 4.9 Applications to Markov Chains

To read: All

Reading Questions

  1. What is the point of studying Markov chains?
  2. What is a steady state vector for a stochastic matrix P?
  3. What is special about regular stochastic matrices?

Submit answers through onCourse

For Tuesday November 3

Section 5.1 Eigenvectors and Eigenvalues
Section 5.2 The Characteristic Equation

To read: All, but you can skim the part of Section 5.2 labeled Determinants. One path through the text is to skip Chapter 3, so that is why this information is included here.

Reading Questions

  1. Let A = . Verify that x = is an eigevector of A with corresponding eigenvalue λ = 3.
  2. Suppose A is 3x3 with eigenvalues 1, 2, and 5. What is the dimension of nul(A)?
  3. Let A be the matrix from #1. Find the characteristic equation of A.

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For Thursday November 5

Section 5.3 Diagonalization

To read: All

Reading Questions

  1. What is the point of finding a diagonalization of a matrix?
  2. If A is 4 x 4 with eigenvalues 1, 2, 0, 3, is A diagonalizable? Explain.

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For Tuesday November 10

Section 5.6 Discrete Dynamical Systems

To read: All

Reading Questions

Consider the discrete dynamical system described by xk+1 = A xk where A is a 2 x 2 matrix.

  1. If the origin is an attractor, what do you know about the eigenvalues of A? Why?
  2. If the origin is a saddle, what do you know about the eigenvalues of A? Why?

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For Thursday November 12

Section 6.1 Inner Product, Length, and Orthogonality

To read: All

Reading Questions

  1. Are the vectors orthogonal in ℝ3? Explain.
  2. Give a geometric interpretation of your answer to 1.
  3. Let H be the yz-plane in ℝ3. What is the orthogonal complement of H?

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For Tuesday November 17

Section 6.2 Orthogonal Sets

To read: All

Reading Questions

  1. Let H be the yz-plane in ℝ3. Give an orthogonal basis for H.
  2. Let w be the orthogonal projection of y onto u. In what direction does w point? In what direction does y - w point?

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For Thursday November 19

Section 6.3 Orthogonal Projections

To read: All

Reading Questions

Let in ℝ3 and let W be the xy-plane in ℝ3.

  1. Find the orthogonal projection of y onto W.
  2. Is there a point in W that is closer to y than your answer in 1? Explain.

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For Tuesday November 24

Exam 3. No Reading Assignment

For Thursday November 26

Thanksgiving Break. No Reading Assignment.

For Tuesday December 1

Section 6.4 The Gram-Schmidt Process

To read: Read the section, but no Reading Questions for today.

For Thursday December 3

Section 6.5 Least-Squares Problems

To read: All

Reading Questions

  1. In your own words, what is the point of this section?
  2. Does every system Ax=b have a least squares solution? Explain.
  3. If a system Ax=b has a least squares solution, must it be unique? Explain.

Submit answers through onCourse

For Tuesday December 8

Section 10.1 Introduction and Examples
Section 10.2 The Steady-State Vector and Google's PageRank

Read the sections, but no Reading Questions for today.

For Thursday December 10

Reread Section 10.1 and 10.2, but no Reading Questions for today.



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