Fall 2001, Math 221

November 2001

Be sure to check back often, because assignments may change!

All section and page numbers refer to sections from Lay, updated 2nd edition.

Due Friday 11/2 at 8am

Section 4.6: Rank
Introduction to Chapter 5
Section 5.1: Eigenvectors and Eigenvalues

• To read: All of Section 4.6 and the Intro to Ch. 5. Through Example 3 in Section 5.1.
• Be sure to understand: In Section 4.6, the definition of rank, the Rank Theorem, and the continuation of the Invertible Matrix Theorem. In Section 5.1, the definitions of eigenvector and eigenvalue, and Example 3.

E-mail Subject Line: Math 221 Your Name 11/2

1. If A is a 4 x 7 matrix with three pivots, what is the dimension of Nul A? Why?
2. Let A= . Verify that (1, -2) is an eigenvector of A with corresponding eigenvalue 3.

Due Monday 11/5 at 8am

Section 5.1: Eigenvectors and Eigenvalues

• Be sure to understand: Example 4

E-mail Subject Line: Math 221 Your Name 11/5

Suppose A is a 3 x 3 matrix with eigenvalues 1, 2, and 5.
1. What is the dimension of Nul(A)?
2. What is the dimension of the eigenspace of A?

Due Wednesday 11/7 at 8am

Section 5.2: The Characteristic Equation

• To read: All. The section on determinants will be review.
• Be sure to understand: The definition of the characteristic equation, Example 3, and the definition of similarity.

E-mail Subject Line: Math 221 Your Name 11/7

1. Let A=. Find the characteristic equation of A.
2. How is the characteristic equation of a matrix related to the eigenvalues of t he matrix?

Due Friday 11/9 at 8am

The Big Picture

• To read: Review Sections 4.5, 4.6, 5.1, 5.2
• Be sure to understand: Try to see the big picture, the connections betweeen the ideas.

E-mail Subject Line: Math 221 Your Name 11/9

1. Write a brief summary of these four sections. As usual, focus on the big ideas and the relationships between them.

Due Monday 11/12 at 8am

Section 5.3: Diagonalization

• Be sure to understand: Example 3

E-mail Subject Line: Math 221 Your Name 11/12

1. What is the point of finding a diagonalization of a matrix?
2. If A is 4x4 with eigenvalues 1, 2, 0, 3, is A diagonalizable? Explain.

Due Wednesday 11/14 at 8am

Section 5.6: Discrete Dynamical Systems

• To read: Through Example 4
• Be sure to understand: Example 1, and the plots in Examples 2, 3 , and 4

E-mail Subject Line: Math 221 Your Name 11/14

1. Let A = . Determine whether the origin is an attractor, a repellor, or a saddle point for solutions to xk+1=Axk.

Reminders:

• You probably don't need reminding, but don't put off working on Exam 2.

Due Friday 11/16 at 8am

Section 6.1: Inner Product, Length, and Orthogonality

• Be sure to understand: The definitions of inner product, norm, and orthogonal complement

Exam 2 due today. No reading questions.

Due Monday 11/19 at 8am

The Big Picture

• To read: Review Sections 5.3, 5.6, 6.1. (You may have already done most of this while taking the exam.)
• Be sure to understand: The big picture, the connections between ideas.

E-mail Subject Line: Math 221 Your Name 11/19

1. Write a brief summary of Section 6.1. Focus on the big ideas, and the connections between them (and the connections between the new ideas and old ones.)

Due Monday 11/26 at 8am

Section 6.2: Orthogonal Sets

• To read: Through the section "Decomposing a Force into Component Forces"
• Be sure to understand: The statement of Theorem 4 and Figure 4

E-mail Subject Line: Math 221 Your Name 11/26

1. Give an example of an orthogonal basis for R3.
2. Let w be the orthogonal projection of y onto u. What direction does w point? What direction does y-w point?

Due Wednesday 11/28 at 8am

Section 6.3: Orthogonal Projections

• To read: Through Example 3
• Be sure to understand: Figure 2 and the statement of the Best Approximation Theorem

E-mail Subject Line: Math 221 Your Name 11/28

Let y=(1, 2, 3) in R3 and let W be the xz-plane.
1. What is the orthogonal projection of y onto W?
2. Is there a point in W that is closer to y than the orthogonal projection you just found? Why or why not?

Due Friday 11/30 at 8am

Due to the problems with my web page and my being behind by a day, this is the same as Wednesday's!)

Section 6.3: Orthogonal Projections

• To read: Through Example 3
• Be sure to understand: Figure 2 and the statement of the Best Approximation Theorem

E-mail Subject Line: Math 221 Your Name 11/28

Let y=(1, 2, 3) in R3 and let W be the xz-plane.
1. What is the orthogonal projection of y onto W?
2. Is there a point in W that is closer to y than the orthogonal projection you just found? Why or why not?

Here ends the reading for November
Go to the reading assignments for December!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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