# Mathematics 330 Homepage

Fall 2019 University of Nevada Reno

330 LINEAR ALGEBRA I (3+0) 3 credits

Vector analysis continued; abstract vector spaces; bases, inner products; projections; orthogonal complements, least squares; linear maps, structure theorems; elementary spectral theory; applications. Corequisite(s): MATH 283 R.

```Instructor  Course Section                      Time
------------------------------------------------------------------------
Eric Olson  Math 330-1006 Linear Algebra        4:30-5:45pm TR AB 634
```

## Course Information

Instructor:
Eric Olson
email:
ejolson at unr edu
Office:
Tuesday and Thursday 2pm DMS 238 and by appointment.
Homepage:
http://fractal.math.unr.edu/~ejolson/330/
Texts:
Linear Algebra and Its Applications, 4th Edition by David C. Lay
https://www.pearson.com/mylab

## Announcements

### [16-Dec-2019] Quiz 3 Question 9

I have scanned my solution of Question 9 from Quiz 3 to help you study for the final exam. Please pay attention to the fact since A ∈ R3×3 and because each of the eigenvalues are of multiplicity one that only two independent rows are needed when computing Nul(A−λI) and that the choice of pivot and free variables can be made to simplify the arithmetic.

### [17-Dec-2019] Final Exam

The final exam will be Tuesday at 2:30 in PSAC 104. It is cumulative and will cover all material from the quizzes plus Sections 6.1 through 6.5. Among other things please be prepared
• State the Gram-Schmidt orthogonalization algorithm.

• Given A ∈ Rm×n, find the factorization A = QR where Q ∈ Rm×n is a matrix with orthonormal columns and R ∈ Rn×n is upper triangular.

• Given the reduced QR factorization of a matrix A, find the vector x which minimizes ||Ax−b||.

### [11-Dec-2019] Homework 5

The fifth computer homework covering chapter 6 and Section 7.1 will be due December 11.

### [05-Dec-2019] Quiz 3

Quiz 3 will be held in class on December 5 covering chapters 1 through 5 with the addition of Section 7.1 but omitting Sections 4.8, 5.7. This quiz is cumulative and may include questions on any topic from Quizzes 1 and 2, their respective study guides. As was as the topics previously discussed, please study the following additional topics to prepare:
• Definitions of the following terms:
• Symmetric matrix.
• Orthogonal matrix.
• Characteristic polynomial.
• Eigenvector and eigenvalue.
• Orthogonally diagonalizable.

• State the following theorems and algorithms:

• Section 7.1 Theorem 3 [The Spectral Theorem]: An n x n symmetric real-valued matrix A has the following properties.
1. A has n real eigenvalues, counting multiplicities.
2. The dimension of the eigenspace for each eigenvalue λ equals the multiplicity of λ as a root of the characteristic equation.
3. The eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding to different eigenvalues are orthogonal.
4. A is orthogonally diagonalizable, that is, there exists an orthonormal basis consisting of eigenvalues of A.

• Section 5.8 [The Power Method for Estimating a Strictly Dominant Eigenvalue]:
1. Select an initial vector x0 whose largest entry is 1.
2. For k = 0, 1, ...,
1. Compute A xk
2. Let μk be an entry in A xk whose absolute value is as large as possible.
3. Compute xk+1 = (1/μk) A xk
3. For almost all choices of x0, the sequence { μk } approaches the dominant eigenvalue, and the sequence { xk } approaches a corresponding eigenvector.

• Know how to prove the following theorems:

• Section 7.1 Theorem 1: If A is real-valued and symmetric, then any two eigenvalues from different eigenspaces are orthogonal.

• Theorem on Real Eigenvalues: The eigenvalues of a real-valued n x n symmetric matrix are all real.

• Compute the characteristic polynomial of a matrix.

• Compute the eigenvalues of a matrix using the characteristic equation.

• Find the eigenvectors of a matrix from the eigenvalues.

### [29-Nov-2019] Homework 4

The fourth computer homework covering Chapters 4 and 5 is due November 29.

### [07-Nov-2019] Quiz 2

Quiz 2 will be held in class on November 7 covering chapters 1 through 4 but omitting Section 4.8. This quiz is cumulative and may include questions on any topic from Quiz 1 and the study guide for Quiz 1. As well as the topics previously discussed, please study the following additional topics to prepare:
• Definitions of the following terms:
• Determinant (using recursive definition).
• Stochastic matrix.

• Proof of the following theorems:
• Section 3.3 Theorem 7 [Cramer's Rule]: Let A be an invertible n×n matrix. For any b in Rn, the unique solution x of Ax=b has entries given by

xi = det(Ai(b))/det(A)   where   i = 1, 2, ..., n.

• Section 4.5 Theorem 9: If a vector space V has a basis B = {b1, ..., bn}, then any set in V containing more than n vectors must be linearly dependent.

• Given a matrix A in Rm×n be able to use Gaussian elimination and pivoting if necessary to find the reduced row-eschelon form.

• Be able to compute det(A) for A in Rn×n given the factorization A=PLDU where L is lower triangular, D is diagonal, U is upper triangular and P is a permutation matrix.

### [01-Nov-2019] Homework 3

The third computer homework covering Chapter 3 is due November 1.

### [03-Oct-2019] Quiz 1

Quiz 1 will be held in class on October 3 covering chapters 1 and 2 from the text. Please study the following topics to prepare:
• Definitions of the following terms:
• A linear function.
• The augmented matrix.
• Row-eschelon and reduced row-eschelon form.
• A subspace.
• The span of a set of vectors span{v1, v2, …, vk}.
• Linear independence of a set of vectors.
• A basis of a subspace.
• The Null space Nul(A) of a matrix.
• The Column space Col(A) of a matrix.

• How to perform the elementary row operations.
• Elimination Step: ri ← ri + αrj.
• Scaling Step: ri ← αri.
• Row Swap: ri ↔ rj.

• Be able to write the matrices which correspond to the elementary row operations.

• Know how to write a system of linear equations as a matrix equation of the form Ax=b and how to write the matrix equation Ax=b as a system of linear equations.

• Be able to interchange the order of elimination and row-swap matrices. For example,

[r2 ← r2 + 3r1] [r1 ↔ r3] = [r1 ↔ r3] [ri ← ri + αrj]

for what values of i, j and α?

• Given a matrix A ∈ Rm×n use the elementary row operations to perform Gaussian elimination and find the row-eschelon and reduced row-eschelon forms of A.

• Given the sequence of row operations used to find the row-eschelon form, find the factorizations

A=LU,   A=LDU   and   A=PLDU

where L is lower triangular, D is diagonal, U is upper triangular and P is a permutation matrix.

• Given a matrix A ∈ Rm×n and it's reduced row-eschelon form R.
• Find a basis for the column space Col(A).
• Find a basis for the null space Nul(A).

### [19-Sep-2019] Homework 2

The first computer homework covering Chapter 2 is due September 9.

### [09-Sep-2019] Homework 1

The first computer homework covering Chapter 1 is due September 9.

### [27-Aug-2019] First Day of Class

Please bring your textbook the first day of class and to all subsequent classes during the semester. To register for the MyLab Math Online portion of this course please consult this information. As refunds from Pearson will not be available if you subsequently drop the course, please select temporary access for now.

```     Quiz 1                    20 points
Quiz 2                    40 points
Quiz 3                    60 points
MyLab Math Online         40 points
Final                    100 points
------------------------------------------
260 points total
```
Exams and quizzes will be interpreted according to the following grading scale:
```    Grade        Minimum Percentage
A                 90 %
B                 80 %
C                 70 %
D                 60 %
```
The instructor reserves the right to give plus or minus grades and higher grades than shown on the scale if he believes they are warranted.

## Quiz and Exam Schedule

There will be three quizzes and a final exam. All quizzes will be held on Thursday's during the usual class meeting time.

## Final Exam

The final exam will be held on Tuesday, December 17 from 2:30 to 4:30pm at PSAC 104. Please note the change in day, time and location from the standard schedule.

## Equal Opportunity Statement

The Mathematics Department is committed to equal opportunity in education for all students, including those with documented physical disabilities or documented learning disabilities. University policy states that it is the responsibility of students with documented disabilities to contact instructors during the first week of each semester to discuss appropriate accommodations to ensure equity in grading, classroom experiences and outside assignments.