Fall 2001, Math 221

September 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 9/7, at 8am

suggestions for reading a math text
course policies
syllabus
Introduction to Chapter 1
Section 1.1: Systems of Linear Equations
Section 1.2: Row Reduction and Echelon Form

• Be sure to understand Example 2 in 1.1; the section "Existence and Uniqueness Questions" in Section 1.2.

E-mail Subject Line: Math 221 Your Name 9/7

Let A be the matrix

1. Is A in row echelon form? Why or why not?
2. What values are in the pivot positions of A?
3. Suppose that A i sthe augmented matrix for a system of 3 equations in 3 unknowns. Is the system consistent or inconsistent? Explain.

Due Monday 9/10 at 8am

Section 1.3 : Vector Equations

• Be sure to understand: The section "Linear Combinations" and the definition of Span{u,v}

E-mail Subject Line: Math 221 Your Name 9/10

1. Let y=(4,9,3), u=(0,1,0), and v=(12,0,9). Write y as a linear combination of u and v.
2. Let u=(1,0,0) and v=(0,1,0). Give a geometric description of Span{u,v}.

Due Wednesday 9/12 at 8am

Section 1.4: The Matrx Equation Ax=b

• Be sure to understand: The statement of Theorem 4

E-mail Subject Line: Math 221 Your Name 9/12

1. Suppose A is a 4x5 matrix with 3 pivots. Do the columns of A span R4?
2. Simplify

Due Friday 9/14 at 8am

Section 1.5: Solution Sets of Linear Systems

• Be sure to understand: Example 3 and the statement of Theorem 6

E-mail Subject Line: Math 221 Your Name 9/14

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?

Due Monday 9/17 at 8am

Linear Independence

• Be sure to understand: The section "Linear Independence of Matrix Columns"

E-mail Subject Line: Math 221 Your Name 9/17

1. If Ax=0 has infinitely many solutions, can the columns of A be linearly independent? Explain.
2. If Ax=b has infinitely many solutions, can the columns of A be linearly independent? Explain.
3. Explain in your own words why a set of three vectors in R2 can not be linearly independent.

Reminder:

• On Monday 9/24 at 5pm, Gregory Rawlins will be giving the Johnson lecture Why Napster Doesn't Matter: Intellectual Property in the Age of Piracy. This is a required part of the course, so make sure it's in your schedule.

Due Wednesday 9/19 at 8am

The Big Picture

• To read: Review Sections 1.1 through 1.6
• Be sure to understand: The big ideas and the deeper connections that you may have missed the first time around

E-mail Subject Line: Math 221 Your Name 9/19

1. Write a brief summary of Sections 1 through 6. It should be short (I don't intend this to take you more than 15 minutes after you finish reviewing), and should focus on the big ideas and the relationships between those ideas.

Due Friday 9/21 at 8am

Section 1.7: Introduction to Linear Transformations

• To read: Through Example 3
• Be sure to understand: Example 1

E-mail Subject Line: Math 221 Your Name 9/21

Let A be the matrix , and let T:R2 --> R2 be defined by T(x)=Ax.
1. Find T(-4,10).
2. Is (4,-2) in the range of T?

Due Monday 9/24 at 8am

Section 1.7: Introduction to Linear Transformations

• Be sure to understand: The definition of a linear transformation

E-mail Subject Line: Math 221 Your Name 9/24

1. Let T:R2 --> R2 be a transformation defined by T(x1, x2)=(x2-3, 4x1+10). Is T a linear transformation?
2. If T:R5 --> R3 is a linear transformation where T(x)=Ax, what is the size of the matrix A?

Reminder:

• Sadly, the Johnson lecture has been postponed indefinitely

Due Wednesday 9/26 at 8am

Section 1.8: The Matrix of a Linear Transformation

• Be sure to understand: Examples 1 and 2, the definition of one-to-one and onto, the statement of Theorems 11 and 12

E-mail Subject Line: Math 221 Your Name 9/26

1. Give the matrix A for the linear transformation T: R2 --> R2 that rotates the plane by Pi/4 degrees counter-clockwise.
2. Give the matrix A for the linear transformation T:R2 --> R2 that expands horizontally by a factor of 2.

Due Friday 9/28 at 8am

Section 1.8: The Matrix of a Linear Transformation

• To read: Re-read it all. Especially focus on the definitions of one-to-one and onto, and on the specific results relating to linear transformations that are one-to-one or onto.
• Be sure to understand: The concepts of one-to-one and onto.

E-mail Subject Line: Math 221 Your Name 9/28

Let T:R5 --> R3 be a linear transformation with standard matrix A, where A has three pivots.
1. Is T one-to-one?

2. Is T onto?

Here ends the reading for September
Go to the reading assignments for October!

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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