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 Friday August 29
Section 1.1 Systems of Linear Equations
Section 1.2 Row Reduction and Echelon Forms
To read:
All
Reading Questions
Let A =
and B =
 Is A in reduced echelon form? Why or why not?
 What are the pivot columns of B? What are the values of the pivots of B?
 If A is the augmented matrix of a linear system, does the system have any free variables?
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For Monday September 1
Labor Day. No Reading Assignment.
For Wednesday September 3
Section 1.3 Vector Equations
To read:
All
Reading Questions
Let u=(1, 3, 2) and v=(2, 1, 2).
 Write the vector w=(1, 11, 2) as a linear
combination of u and v.
 Give a geometric description of Span{u, v}.
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For Friday September 5
Section 1.4 The Matrix Equation Ax=b
To read:
All
Reading Questions
Let A =
 If b = , does Ax = b have a solution? Why or why not?
 Does Ax = b have a solution for every b ∈ ℝ^{3}? Why or why not?
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For Monday September 8
Section 1.5 Solution Sets of Linear Systems
To read:
All
Reading Questions
 Explain the difference between a homogeneous system of equations and a nonhomogeneous system of equations.
 If the system Ax = b is consistent and Ax = 0 has a
nontrivial solution, how many solutions does Ax =b have?
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For Wednesday September 10
Section 1.7 Linear Independence
To read:
All
Reading Questions
Let
 Is the set of vectors {u, v} linearly independent or linearly dependent?
Explain.
 Is the set of vectors {u, v, w} linearly independent or linearly dependent? Explain.
 If the vectors u, v, w, and z form the columns of the matrix A, does Ax = 0 have a nontrivial solution? Explain.
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For Friday September 12
Section 1.8 Introduction to Linear Transformations
To read:
All
Reading Questions
 If T: ℝ^{3} → ℝ^{6} and T(x) = Ax
for some matrix A, what are the dimensions of A? Why?
 Give an example of a function T: ℝ^{2} → ℝ^{2}
that is not a linear transformation. Explain.
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For Monday September 15
Section 1.9 The Matrix of a Linear Transformation
To read:
All
Reading Questions
 Give the matrix A for the linear transformation
T:ℝ^{2} → ℝ^{2} that expands horizontally by a factor of 3.
 Let T:ℝ^{5} → ℝ^{3} be a linear transformation with
standard matrix A where A has three pivots. Is T oneone? Explain.
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For Wednesday September 17
Flex Day. No Reading Assignment.
For Friday September 19
Section 2.1 Matrix Operations
To read:
All
Reading Questions
 Give one way in which matrix multiplication differs from multiplication of real numbers.
 Let
Compute the following products or explain why it is impossible:
AB, BA, BA^{T}
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For Monday September 22
Exam 1 tonight. No Reading Assignment.
For Wednesday September 24
Section 2.2 Inverse of a Matrix
To read:
All
Reading Questions
 Let
. Find B^{1}.
 Give an example of a 2 x 2 singular matrix.
 If A is invertible, how many solutions does Ax=b have? Why?
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For Friday September 26
Section 2.3 Characterizations of Invertible Matrices
To read:
All
Reading Questions
 If A is an n x n matrix with n pivots, how many solutions does Ax=b have? Why?
 If A is an invertible matrix, are the rows of A a linearly independent set or a linearly dependent set? Why?
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For Monday September 29
Section 2.7 Applications to Computer Graphics
To read:
All
Reading Question
What is the advantage of using homogeneous coordinates in computer graphics?
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For Wednesday October 1
Section 3.1 Introduction to Determinants
To read:
All
Reading Questions
 Let
. Find det(A).
 Let
. Find det(B).
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For Friday October 3
Section 3.2 Properties of Determinants
To read:
All
Reading Question
Why do we care about finding det(A)?
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For Monday October 6
Section 4.1 Vector Spaces and Subspaces
To read:
All
Reading Questions
 Give an example of a subset of ℝ^{2} that is not a subspace
of ℝ^{2}.
 Let ℙ_{4} denote the set of all polynomials of degree 4 or less with real coefficients. Give examples of two vectors in the vector space ℙ_{4}.
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For Wednesday October 8
Section 4.2 Null Spaces, Column Spaces, and Linear Transformations
To read:
All
Reading Questions
 If the columns of A are linearly independent, what is Nul(A)? Why?
 If A is m x n with m pivots, what is Col(A)? Why?
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For Friday October 10
Section 4.3 Linearly Independent Sets; Bases
To read:
All
Reading Questions
Let
 Do the columns of A form a basis for ℝ^{3}? Explain.
 Give a basis for Col(A).
 Give a basis for Nul(A).
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For Monday October 13
Fall Break. No Reading Assignment.
For Wednesday October 15
Section 4.3 Linearly Independent Sets; Bases
Reread the section, but no Reading Questions for today.
For Friday October 17
Section 4.4 Coordinate Systems
To read:
All
Reading Questions
Let
 Show that B is a basis for ℝ^{2}.
 Find the coordinate vector [x]_{B} of x relative to B.
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For Monday October 20
Exam 2 tonight. No Reading Assignment.
For Wednesday October 22
Section 4.5 The Dimension of a Vector Space
To read:
All
Reading Questions
 What is the dimension of ℝ^{3}? Why? Does this make sense geometrically?
 Is there be a set of linearly independent vectors {v_{1}, . . ., v_{12}} that does not span ℝ^{12}? Explain.
 If A is 4x7 with three pivots, what is the dimension of Nul(A)? Why?
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For Friday October 24
Section 4.6 Rank
To read:
All
Reading Questions
 If A is 4x7 with three pivots, what is the dimension of row(A)? Why?
 Let A be the matrix from the reading for October 10. Give a basis for
the row space of A. Explain.
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For Monday October 27
Section 4.9 Applications to Markov Chains
To read:
All
Reading Questions
 What is the point of studying Markov chains?
 What is a steady state vector for a stochastic matrix P?
 What is special about regular stochastic matrices?
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For Wednesday October 29
Section 5.1 Eigenvectors and Eigenvalues
To read:
All
Reading Questions
 Let A = .
Verify that x =
is an eigevector of A with corresponding eigenvalue λ = 3.
 Suppose A is 3x3 with eigenvalues 1, 2, and 5. What is the dimension of nul(A)?
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For Friday October 31
Section 5.2 The Characteristic Equation
To read:
All
Reading Questions
 Let A be the matrix from the reading for October 29. Find the characteristic equation of A.
 How is the characteristic equation of a matrix related to the eigenvalues of the matrix?
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For Monday November 3
Flex day. No Reading Assignment.
For Wednesday November 5
Section 5.3 Diagonalization
To read:
All
Reading Questions
 What is the point of finding a diagonalization of a matrix?
 If A is 4 x 4 with eigenvalues 1, 2, 0, 3, is A diagonalizable? Explain.
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For Friday November 7
Section 5.6 Discrete Dynamical Systems
To read:
All
Reading Questions
Consider the discrete dynamical system described by x_{k+1} = A x_{k} where A is a 2 x 2 matrix.
 If the origin is an attractor, what do you know about the eigenvalues of A? Why?
 If the origin is a saddle, what do you know about the eigenvalues of A? Why?
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For Monday November 10
Section 6.1 Inner Product, Length, and Orthogonality
To read:
All
Reading Questions
 Are the vectors
orthogonal in ℝ^{3}? Explain.
 Give a geometric interpretation of your answer to 1.
 Let H be the xzplane in ℝ^{3}. What is the orthogonal complement of H?
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For Wednesday November 12
Section 6.1 Inner Product, Length, and Orthogonality
Reread the section, but no Reading Questions for today.
For Friday November 14
Section 6.2 Orthogonal Sets
To read:
All
Reading Questions
 Let H be the xzplane in ℝ^{3}. Give an orthogonal basis for H.
 Let w be the orthogonal projection of y onto u. What direction does w point? What direction does y  w point?
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For Monday November 17
Exam 3 tonight. No Reading Assignment.
For Wednesday November 19
Section 6.3 Orthogonal Projections
To read:
All
Reading Questions
Let
in ℝ^{3} and let W be the xyplane in ℝ^{3}.
 Find the orthogonal projection of y onto W.
 Is there a point in W that is closer to y than your answer in 1? Explain.
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For Friday November 21
No class meeting today. No Reading Assignment.
For Monday November 24
Section 6.5 LeastSquares Problems
To read:
All
Reading Questions
 In your own words, what is the point of this section?
 Does every system Ax=b have a least squares solution? Explain.
 If a system Ax=b has a least squares solution, must it be unique? Explain.
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For Wednesday November 26
Thanksgiving Break. No Reading Assignment.
For Friday November 28
Thanksgiving Break. No Reading Assignment.
For Monday December 1
Section 6.6 Applications to Linear Models
To read:
All
Reading Questions
 In your own words, what is the point of this section?
 What is the connection between Example 2 from this section and Problem 34 from Section 1.2? Explain.
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