## MTH 266 - Linear Algebra at Piedmont Virginia Community College

### Course Description

Effective: 2017-08-01

Covers matrices, vector spaces, determinants, solutions of systems of linear equations, basis and dimension, eigenvalues, and eigenvectors. Features instruction for mathematical, physical and engineering science programs.

Lecture 3 hours. Total 3 hours per week.

3 credits

### General Course Purpose

The general purpose is to give the student a solid grasp of the methods and applications of linear algebra, and to prepare the student for further coursework in mathematics, engineering, computer science and the sciences.

### Course Prerequisites/Corequisites

Prerequisite: Completion of MTH 263 or equivalent with a grade of B or better or MTH 264 or equivalent with a grade of C or better.

### Course Objectives

- Matrices and Systems of Equations
- Use correct matrix terminology to describes various types and features of matrices (triangular, symmetric, row echelon form, et.al.)
- Use Gauss-Jordan elimination to transform a matrix into reduced row echelon form
- Determine conditions such that a given system of equations will have no solution, exactly one solution, or infinitely many solutions
- Write the solution set for a system of linear equations by interpreting the reduced row echelon form of the augmented matrix, including expressing infinitely many solutions in terms of free parameters
- Write and solve a system of equations modeling real world situations such as electric circuits or traffic flow
- Matrix Operations and Matrix Inverses
- Perform the operations of matrix-matrix addition, scalar-matrix multiplication, and matrix-matrix multiplication on real and complex valued matrices
- State and prove the algebraic properties of matrix operations
- Find the transpose of a real valued matrix and the conjugate transpose of a complex valued matrix
- Identify if a matrix is symmetric (real valued)
- Find the inverse of a matrix, if it exists, and know conditions for invertibility.
- Use inverses to solve a linear system of equations
- Determinants
- Compute the determinant of a square matrix using cofactor expansion
- State, prove, and apply determinant properties, including determinant of a product, inverse, transpose, and diagonal matrix
- Use the determinant to determine whether a matrix is singular or nonsingular
- Use the determinant of a coefficient matrix to determine whether a system of equations has a unique solution
- Norm, Inner Product, and Vector Spaces
- Perform operations (addition, scalar multiplication, dot product) on vectors in Rn and interpret in terms of the underlying geometry
- Determine whether a given set with defined operations is a vector space
- Basis, Dimension, and Subspaces
- Determine whether a vector is a linear combination of a given set; express a vector as a linear combination of a given set of vectors
- Determine whether a set of vectors is linearly dependent or independent
- Determine bases for and dimension of vector spaces/subspaces and give the dimension of the space
- Prove or disprove that a given subset is a subspace of Rn
- Reduce a spanning set of vectors to a basis
- Extend a linearly independent set of vectors to a basis
- Find a basis for the column space or row space and the rank of a matrix
- Make determinations concerning independence, spanning, basis, dimension, orthogonality and orthonormality with regards to vector spaces
- Linear Transformations
- Use matrix transformations to perform rotations, reflections, and dilations in Rn
- Verify whether a transformation is linear
- Perform operations on linear transformations including sum, difference and composition
- Identify whether a linear transformation is one-to-one and/or onto and whether it has an inverse
- Find the matrix corresponding to a given linear transformation T: Rn -> Rm
- Find the kernel and range of a linear transformation
- State and apply the rank-nullity theorem
- Compute the change of basis matrix needed to express a given vector as the coordinate vector with respect to a given basis
- Eigenvalues and Eigenvectors
- Calculate the eigenvalues of a square matrix, including complex eigenvalues.
- Calculate the eigenvectors that correspond to a given eigenvalue, including complex eigenvalues and eigenvectors.
- Compute singular values
- Determine if a matrix is diagonalizable
- Diagonalize a matrix

### Major Topics to be Included

- Matrices and Systems of Equations
- Matrix Operations and Matrix Inverses
- Determinants
- Norm, Inner Product, and Vector Spaces
- Basis, Dimension, and Subspaces
- Linear Transformations
- Eigenvalues and Eigenvectors