MTH 2340 - Linear Algebra with Applications

• Perform a sequence of elementary row operations to find the row echelon and reduced row echelon form of a matrix
• Identify the pivot positions and rank of a matrix
• Determine if a system of linear equations is consistent or inconsistent by considering a row echelon form of its augmented matrix
• Describe the solution set of a consistent system of linear equations, including identifying any free variables
• Use the terminology associated with linear combinations and spanning sets
• Perform matrix-vector multiplication and interpret the result as a linear combination
• Associate a system of linear equations with a vector equation or matrix equation, and vice versa
• Express the solutions of a consistent matrix equation in parametric vector form
• Determine if a set of vectors is linearly independent
• Find non-trivial linear dependence relations among a set of linearly dependent vectors
• Use the terminology associated with linear transformations, including domain, codomain, range, onto, and one-to-one
• Prove that a given transformation is or is not a linear transformation by checking the conditions of the definition
• Determine the matrix of a linear transformation, either from its formula or from a geometric description
• Perform matrix-matrix multiplication
• Find the transpose of a matrix
• Determine if a matrix is invertible and find its inverse using row reduction
• Apply the invertible matrix theorem to draw connections between topics in linear algebra
• Prove that a given set of vectors is or is not a subspace of Rⁿ
• Determine if a given set of a vectors is a basis for a given subspace of Rⁿ
• Determine the dimension of a subspace of Rⁿ
• Find a basis and dimension for the null space and column space of a matrix
• Apply properties of the determinant
• Find the determinant of a matrix via row reduction and cofactor expansion
• Find the determinant of a triangular or diagonal matrix
• Interpret the determinant of a 2x2 or 3x3 invertible matrix geometrically
• Show that a real number is an eigenvalue of a matrix with real number entries by finding the corresponding eigenvectors
• Determine the eigenvalues of a matrix and their algebraic multiplicity using the characteristic equation
• Diagonalize a matrix, if possible, or determine it is not diagonalizable
• Employ eigenvectors and eigenvalues in selected applications and interpret the results
• Use the inner product to find the length of a vector and the distance between two vectors
• Determine if a set of vectors is an orthogonal basis for a subspace of Rⁿ
• Find the projection of a vector onto a subspace of Rⁿ

• Elementary geometry
• Cartesian coordinates
• Linear equations
• Vector algebra in 2D and 3D

• Systems of linear equations, elementary row operations, and echelon form
• Linear combinations and spanning sets
• Matrix equations
• Linear independence and dependence
• Linear transformations and their matrices
• Matrix algebra and invertible matrices
• Subspaces, bases, and dimension
• The determinant and its properties
• Eigenvectors and eigenvalues, diagonalization, application to general linear models
• Inner products, orthogonality, projection, and selected applications
• The spectral theorem, singular value decomposition, and selected applications