### Reading Assignments - Math 221 Linear Algebra - Fall 2014

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

Let A = and B =

1. Is A in reduced echelon form? Why or why not?
2. What are the pivot columns of B? What are the values of the pivots of B?
3. 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

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

1. Write the vector w=(-1, 11, 2) as a linear combination of u and v.
2. Give a geometric description of Span{u, v}.

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#### For Friday September 5

Section 1.4 The Matrix Equation Ax=b

Let A =

1. If b = , does Ax = b have a solution? Why or why not?
2. Does Ax = b have a solution for every b ∈ ℝ3? Why or why not?

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#### For Monday September 8

Section 1.5 Solution Sets of Linear Systems

1. Explain the difference between a homogeneous system of equations and a non-homogeneous system of equations.
2. If the system Ax = b is consistent and Ax = 0 has a non-trivial solution, how many solutions does Ax =b have?

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#### For Wednesday September 10

Section 1.7 Linear Independence

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 Friday September 12

Section 1.8 Introduction to Linear Transformations

1. If T: ℝ3 → ℝ6 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 Monday September 15

Section 1.9 The Matrix of a Linear Transformation

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

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#### For Wednesday September 17

Flex Day. No Reading Assignment.

#### For Friday September 19

Section 2.1 Matrix Operations

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

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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

1. Let . Find B-1.
2. Give an example of a 2 x 2 singular matrix.
3. 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

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?

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#### For Monday September 29

Section 2.7 Applications to Computer Graphics

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

1. Let . Find det(A).

2. Let . Find det(B).

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#### For Friday October 3

Section 3.2 Properties of Determinants

Why do we care about finding det(A)?

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#### For Monday October 6

Section 4.1 Vector Spaces and Subspaces

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.

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#### For Wednesday October 8

Section 4.2 Null Spaces, Column Spaces, and Linear Transformations

1. If the columns of A are linearly independent, what is Nul(A)? Why?
2. 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

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 Monday October 13

Fall Break. No Reading Assignment.

#### For Wednesday October 15

Section 4.3 Linearly Independent Sets; Bases

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

#### For Friday October 17

Section 4.4 Coordinate Systems

1. Show that B is a basis for ℝ2.
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

1. What is the dimension of ℝ3? Why? Does this make sense geometrically?
2. Is there be a set of linearly independent vectors {v1, . . ., v12} that does not span ℝ12? Explain.
3. 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

1. If A is 4x7 with three pivots, what is the dimension of row(A)? Why?
2. 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

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?

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#### For Wednesday October 29

Section 5.1 Eigenvectors and Eigenvalues

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)?

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#### For Friday October 31

Section 5.2 The Characteristic Equation

1. Let A be the matrix from the reading for October 29. Find the characteristic equation of A.
2. 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

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 Friday November 7

Section 5.6 Discrete Dynamical Systems

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 Monday November 10

Section 6.1 Inner Product, Length, and Orthogonality

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

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#### For Wednesday November 12

Section 6.1 Inner Product, Length, and Orthogonality

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

#### For Friday November 14

Section 6.2 Orthogonal Sets

1. Let H be the xz-plane in ℝ3. Give an orthogonal basis for H.
2. 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

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 Friday November 21

No class meeting today. No Reading Assignment.

#### For Monday November 24

Section 6.5 Least-Squares Problems

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.

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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