M341 LINEAR ALGEBRA AND MATRIX THEORY
Prerequisite and degree relevance: The prerequisite is one of M408D, M408L, M408S or the equivalent, with a grade of at least C-, or consent of instructor. (Credit may not be received for both M341 and M340L. Majors with a 'math' advising code must register for M341 rather than for M340L; majors without a 'math' advising code must register for M340L. Math majors must make a grade of at least C- in M341.)
Course description: The emphasis in this course is on understanding the concepts and learning to use the tools of linear algebra and matrices. Some time should be devoted to teaching students to do proofs. The fundamental concepts and tools of the subject covered are:
- Matrices: matrix operations, the rules of matrix algebra, invertible matrices.
- Linear equations: row operations and row equivalence; elementary matrices; solving ystems of linear equations by Gaussian elimination; inverting a matrix with the aid of row operations.
- Vector spaces:vector spaces and subspaces; linear independence and span of a set of vectors, basis and dimension; the standard bases for common vector spaces.
- Inner product spaces: Cauchy-Schwarz inequality, orthonormal bases, the Gramm-Schmidt procedure, orthogonal complement of a subspace, orthogonal projection.
- Linear Transformations: kernel and range of a linear transformation, the Rank- Nullity Theorem, linear transformations and matrices, change of basis, similarity of matrices.
- Determinants: the definition and basic properties of determinants, Cramers rule.
- Eigenvalues: eigenvalues and eigenvectors, diagonalizability of a real symmetric matrix, canonical forms.