# Exam #1 Review

## Scheduling Information:

Time: Friday, September 28, during class time.
Location: McConomy Auditorium.
Review Session: Wednesday, September 26, from 7:00-8:30pm in POS A35 (Mellon Aud).

## Topics Covered:

The Fundamental Problem in Linear Algebra: Section 2.1.
Solving Systems of Linear Equations: Section 2.2.
Matrices and Gaussian Elimination: Sections 2.1 and 2.2.
Logic. What is a Proof?: Introduction to Mathematical Arguments.
What is a Theorem? Techniques of proof. Introduction to Mathematical Arguments.
Vector Spaces: Sections 1.1 and 6.1.
Geometry of Linear Equations: Sections 1.2 and 1.3.
Linear Transformations: Section 3.6 pp. 211-214, and Section 6.4 all
Matrix Multiplication: Section 3.1 pp. 138-145.
Matrix Algebra: Section 3.1 pp 144-151 and 155-158, and Section 3.2 all.

## Review Questions:

1. What is a linear system of equations?
2. What are the three operations we can perform on a system?
3. How many solutions might an system have?
4. What are the leading variables of a system? The free variables? How do we decide which are which?
5. What is a coefficient matrix? Augmented matrix?
6. What are the elementary row operations?
7. What is upper triangular form?
8. What is row echelon form?
9. What is reduced row echelon form?
10. Can you give an example of an upper triangular matrix that is not in row echelon form? Row echelon form, but not reduced row echelon form?
11. What is a "statement?"
12. What are the five logical operators we discussed? How do they work?
13. What are the two qualifiers we discussed? How do they work?
14. How do you negate statements involving the five operators we discussed?
15. How do you negate statements involving the two quantifiers we discussed?
16. How do you negate more complicated expressions?
17. What two types of theorems did we discuss?
18. How can you prove P->Q?
19. What is the most important thing to keep in mind when constructing a proof using cases?
20. What is a vector space? What are the properties of a vector space?
21. Why are those properties true in R^n?
22. What is the row picture of a system?
23. What is the column picture of a system?
24. What is the Vector Form for a system?
25. What do I mean when I say "solve" a system?
26. What is a real vector space? A complex vector space?
27. What do linear transformations have to do with linear combinations
28. Why is differentiation a linear transformation?
29. What are two linear transformations that can be defined for any vector space?
30. What are the standard basis vectors?
31. Why does the action of a linear transformation on the standard basis vectors completely determine the linear transformation? What does that mean?
32. What is the product of a matrix and a vector? What restrictions are there on this multiplication?
33. What is the composition of linear transformations?
34. What is the transpose of a matrix?
35. What is the sum of two matrices? When is it defined?
36. What is the scalar product of a matrix?
37. How do you multiply [matrix]x[vector]? What are the two representations we have for that product?
38. What is the product of two matrices?

## Exercises:

Section 2.1: #28, 30, 34, 36, 39.
Section 2.2 #1, 3, 5, 9, 11, 13, 27, 29, 31.
Introduction to Mathematical Arguments: Section 1.3 #2, 4.
Section 1.1: #7, 9, 13, 15, 23, prove theorem 1.1 (a,b in text, e done in class).
Section 6.1: #5, 7, 13, 17.
Section 3.6: #1, 5, 9, 31, 33.
Section 6.4: #9, 11, 15, 17, 27.
Section 3.1 #1, 5, 9, 13, 21.
Section 3.2 #5, 17, 19.

## Old Exam Problems:

Here are some old exam problems that I have written and given to 21-241 students in previous semesters.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. You'll have a chance to ask questions at the Review Session, and during your TA's office hours, too.