Course Content

• Definition of vectors
• Systems of linear equations and their solution sets
• Homogeneous linear systems

• Augmented matrix for a systems of equations
• Elementary row operations and row echelon form
• Solving systems of equations using Gaussian Elimination
• Using Gauss-Jordan Reduction to put an augmented matrix in reduced row echelon form

• Symmetric Matrices, the transpose, and the trace
• Linear combinations of matrices
• Matrix multiplication and its properties
• The inverse of a matrix and its properties

• Properties of elementary matrices
• Finding the inverse of a matrix
• Properties of invertible matrices
• The determinant of a matrix

• Calculating the determinant of a matrix
• Properties of the determinant
• How row operations change the determinant
• The determinant of invertible and noninvertible matrices

• Cramer's Rule
• The adjoint and the inverse of a matrix
• The definition and properties of vector spaces
• Subspaces of a vector space

• The span of a subset and its properties
• The definition and properties of linear independence
• Basis for a vector space

• Determining if a set is a basis for a vector space
• Properties of a basis for a vector space
• The dimension of a vector space
• The column, row, and null spaces of a matrix
• Rank and nullity of a matrix
• Coordinates of a vector space

• Using a basis of a vector space to create an isomorphism to \(\mathbb{R}^n\)
• Finding a basis of a subspace from a spanning set
• Change of basis

• Rank-Nullity Theorem
• Linear transformations
• Kernel and range of linear transformations
• Definition of an isomorphism

• Properties of linear transformations including injective, surjective, and bijective
• Matrix representations of linear transformations between vector spaces
• The relationship between the matrix for a linear transformation and the range and kernel of the linear transformation
• Rank-Nullity Theorem (or Dimension Theorem) for linear transformations

• Examples of linear transformations and the application of the Rank-Nullity Theorem
• Relationship between transition matrices and linear transformations
• Using transition matrices to change the matrix representation of a linear representation to use a different basis of the vector space
• Similarity of matrices
• Eigenvalues and eigenvectors

• Properties of eigenvalues and eigenvectors
• Definition of the eigenspace of an eigenvalue 
• Characteristic polynomial of a matrix
• Algebraic and geometric multiplicity of an eigenvalue
• Determining if a matrix is diagonalizable
• Vector spaces with norms