Mathematics 330 Homepage

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
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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

[17-Dec-2019] Final Exam

• State the Gram-Schmidt orthogonalization algorithm.

• Given A ∈ R^m×n, find the factorization A = QR where Q ∈ R^m×n is a matrix with orthonormal columns and R ∈ R^n×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

[05-Dec-2019] Quiz 3

• 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 x_k
       2. Let μ_k be an entry in A x_k whose absolute value is as large as possible.
       3. Compute x_k+1 = (1/μ_k) A x_k
    3. For almost all choices of x₀, the sequence { μ_k } approaches the dominant eigenvalue, and the sequence { x_k } 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

[07-Nov-2019] Quiz 2

• 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
    x_i = det(A_i(b))/det(A)   where   i = 1, 2, ..., n.

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

• Given a matrix A in R^m×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 R^n×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

[03-Oct-2019] Quiz 1

• 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, …, v_k}.
  • 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: r_i ← r_i + αr_j.
  • Scaling Step: r_i ← αr_i.
  • Row Swap: r_i ↔ r_j.

• 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] [r_i ← r_i + αr_j]

  for what values of i, j and α?

• Given a matrix A ∈ R^m×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 ∈ R^m×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

[09-Sep-2019] Homework 1

[27-Aug-2019] First Day of Class

Grading

     Quiz 1                    20 points
     Quiz 2                    40 points
     Quiz 3                    60 points
     MyLab Math Online         40 points
     Final                    100 points
    ------------------------------------------
                              260 points total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Quiz and Exam Schedule

Final Exam

Equal Opportunity Statement

Academic Conduct