MATH 4435/6435 – Content Standards
Title: Linear Algebra
Prerequisite: Math 3435 with grade of C or higher.
Matrix representations of linear transformations, similarity; inner product spaces and orthogonality, least squares problems; eigenvalues, diagonalization, Hermitian matrices, quadratic forms, positive definite matrices, and nonnegative matrices.
Goals
To broaden and deepen students’ understanding of linear algebra acquired from an introductory course, and to adequately prepare students for applying linear algebra in other areas or subjects and for pursuing advanced graduate courses in matrix theory.
Content Standards
The following standards are offered as guidelines for assessing student progress, judging the effectiveness of instructional programs, and developing curricular units. The subject matter outlined in these standards represents the minimum knowledge in which a student should demonstrate proficiency at the successful completion of the course.
Students will be able to recognize a linear transformation and to find the kernel and range of a linear transformation.
Students will be able to compute the matrix representation of a linear transformation with respect to given bases.
Students will be able to derive properties of a linear transformation if a matrix representation of it is known.
Students will be able to find the transition matrix from a basis to another basis.
Students will be able to use the similarity between matrices representing the same linear operator from a vector space into itself under difference bases.
Students will be able to show that similar matrices share many common properties.
Students will be able to find the norm of a vector and the angle between two nonzero vectors using an inner product.
Students will be able to state, prove and apply the Cauchy-Schwarz Inequality.
Students will be able to find the distance between a point and a line (or plane) using orthogonal projection.
Students will be able to compute the orthogonal complement of a subspace.
Students will be able to prove and apply the Fundamental Subspace Theorem for matrices.
Students will be able to show and utilize the fact that an inner product space can be written as the direct sum of any subspace and its orthogonal complement.
Students will be able to find the least squares solutions to a linear system by solving the normal equations.
Students will be able to find the best least squares fit by a linear (or quadratic) function to given data.
Students will be able to recall some typical inner product spaces, such as the vector space of real continuous functions on a closed interval and the vector space of all polynomials in x of degree less than a fixed number.
Students will be able to compute various norms that can be defined in a vector space.
Students will be able to demonstrate that every orthogonal set of nonzero vectors is linearly independent.
Students will be able to use properties of orthonormal bases to compute the inner product of two vectors.
Students will be able to establish several conditions on a matrix Q each of which is equivalent to Q being an orthogonal matrix.
Students will be able to compute the orthogonal projection of a vector onto a subspace S if an orthonormal basis for S is known.
Students will be able to find the best least squares approximation to a given function on a closed interval [a, b] by a linear (or quadratic) function.
Students will be able to find the projection matrix onto a subspace of the Euclidean space of dimension n.
Students will be able to apply the Gram-Schmidt orthogonalization process to find an orthonormal basis for (a subspace of) an inner product space.
Students will be able to compute the QR factorization of a matrix
Students will be able to compute the eigenvalues and eigenvectors of a matrix (of small order).
Students will be familiar with some identities involving the product and sum of the eigenvalues of a matrix.
Students will be able to solve certain linear systems of differential equations using eigenvalues and eigenvectors of the coefficient matrix, or using the matrix exponential.
Students will be able to determine if a given matrix is diagonalizable.
Students will be able to find a matrix that diagonalizes a given diagonalizable matrix A and use this to compute a power of A and the matrix exponential of A.
Students will be able to define and compute complex inner products.
Students will be able to prove Schur’s (Upper Triangularization) Theorem.
Students will be able to show that the eigenvalues of a Hermitian matrix H are real and H is unitarily diagonalizable.
Students will be able to demonstrate that a matrix is normal if and only if it is unitarily diagonalizable.
[Optional] Students will be able to compute and apply a singular value decomposition of a matrix.
Students will be able to determine if a real symmetric matrix (or a quadratic form) is positive definite, or positive semidefinite.
Students will be able to establish the equivalence of several conditions each of which is equivalent to the matrix A being positive definite.
[Optional] Students will be able to state and apply the Perron-Frobenius theorems on nonnegative matrices.