MTH 266 Linear Algebra (3 cr.)
Covers matrices, vector spaces, determinants, solutions of systems of linear equations, basis and dimension, eigenvalues, and eigenvectors. Designed for mathematical, physical and engineering science programs. This course replaces MTH 177 or MTH 275 or MTH 285. 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. Lecture 3 hours per week.
Major Topics:
Matrices and Systems of Equations
Matrix Operations and Matrix Inverses
Determinants and Eigenvalues
Norm, Inner Product, and Vector Spaces
Basis, Dimension, and Subspaces
Linear Transformations
Eigenvalues and Eigenvectors
Objectives and Skills:
Use correct matrix terminology to describe various types and features of matrices (e.g., triangular, symmetric, row echelon)
Use Gauss-Jordan elimination to transform a matrix into reduced row echelon form
Determine conditions when a given system of equations will have no solution, exactly one solution, or infinitely many solutions
Write 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
Perform the operations of matrix addition, scalar multiplication, and matrix multiplication with 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 real-valued matrix is symmetric
Find the inverse of a matrix, if it exists, and determine conditions for invertibility
Use inverses to solve a linear system of equations
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
Perform operations (addition, scalar multiplication, dot product) on vectors in real-valued n-dimensional space
Determine whether a given set with defined operations is a vector space
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 dimensions of vector spaces and subspaces
Prove or disprove that a given subset is a subspace of real-valued n-dimensional space
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 regard to vector spaces
Use matrix transformations to perform rotations, reflections, and dilations in real-valued n-dimensional space
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 (injective) or onto (surjective), 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
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