# Math 333 - Fall 2005

**TIME:**TR 8-9:15am (Wilson 1-117)**TEXTBOOK:**Elementary Linear Algebra, H. Anton, C. Rorres (9th edition)**INSTRUCTOR:**Mark Pernarowski**OFFICE:**Wilson 2-236**OFFICE HOURS:**Schedule M,F 9-10; Tu 10-11, W 12-1**PHONE:**994-5356-
**GRADING SYSTEM:**

The course % will determined as follows:Total Points Midterm M 100 Final F 100 Homework HW 200 Course % = (M+F+HW)/4

Homework due dates and the date for the midterm will be announced in class. Exact content of the exams will also be announced in class. The final exam is NOT comprehensive. There is no set schedule for when homework will be assigned. Some assignments will be longer than others. As a result, the HW grade above will be the percentage obtained.

**Topics Covered:**

- Vectors in R^n
- Axiomatic Definition
- Dot Product, Norm, Distance

- Linear Transformations from R^n to R^m
- Reflection, Rotation, Projection
- Generating matrices from standard basis
- Inverse, 1-1, equivalent conditions

- General Vector Spaces
- Axiomatic Definitions
- Examples of different addition and scalar multiplication
- R^n, P_n, M_{nn}, and C^n(R) function spaces
- Subspace Theorem
- Spanning sets, Linear Independence, Basis
- Plus/Minus Theorems
- Basis, dimension Theorems
- Coordinates relative to a basis
- Finding bases in coordinate spaces

- Fundamental Matrix Spaces
- row(A), col(A), N(A), N(A^T)
- Relation to coordinate spaces of subspaces

- Inner Product Spaces
- Real inner product spaces, norm, distance
- Cauchy-Schwartz inequality
- Orthogonality, orthogonal bases
- Gram-Schmidt process
- Projections onto subspaces
- Orthogonal Complement Spaces
- Sum and Direct sum
- Orthogonal Decomposition of Finite Dimensional Spaces
- Orthogonality of Fundamental Matrix Spaces
- Relation to Coordinate Spaces

- Eigenvalues and Eigenvectors
- Definition, Characteristic Polynomials
- Algebraic and Geometric Multiplicity of Eigenspaces
- Diagonalizability Theorems
- Symmetric Matrices, Orthogonal Diagonalization
- Functions of Symmetric Matrices, Iterations

- Linear Transformations
- Definitions and Examples
- Images from basis vectors, Compositions
- Kernel, Range
- Generating matrices and commuting diagrams in coordinate space
- Inverse transformations, linearity
- Eigenfunctions, values for general transformations
- Differential operator examples

**Homework Assignments**

- Homework 1 (Due Tuesday, Sept. 13 in class) (PDF File)
- Homework 2 (Due Thursday, Sept. 22 in class) (PDF File)
- Homework 3 (Due Tuesday, Oct. 4 in class) (PDF File)
- Homework 4 (Due Thursday, Oct. 13 in class) (PDF File)
- Midterm 1 (Nov 20)
- Homework 5 (Due Tuesday, Nov. 3 in class) (PDF File)
- Homework 6 (Due Tuesday, Nov. 15 in class) (PDF File)
- Homework 7 (Due Tuesday, Nov. 22 in class) (PDF File)
- Homework 8 (Due Tuesday, Dec. 6 in class) (PDF File)

**Notes (in progress)**

- Course Outline (PDF File)
- Summary Notes (PDF File) Summarizes basic definitions, Theorems on vector spaces, spanning, independence and bases. Also, includes fundamental matrix spaces, their bases and related theorems.
- File of symbol definitions (PDF File)