MATH 251(2004): Topics and Main Theorems Covered
The following topics will be covered by the end of the course.
Chapter references are to the text (2nd edition). You should know the definition of all terms and have examples of each.
- Ch 3 Matrices: all
- Ch 4 Square Matrices: all
- Ch 5 Vector Spaces: all
- Ch 6 Inner Product Spaces, Orthogonality: all
- Ch 8 Eigenvectors and Eigenvalues: all
- Ch 9 Linear Mappings: all
- Ch 10 Matrices and Linear Mappings: all
- Ch 11 Canonical Forms: all except for the rational canonical form
- Ch 12 Linear Functionals and the Dual Space
- Ch 13 Bilinear, Quadratic and Hermitian forms: all
- Ch 14 Linear operators on Inner Product Spaces: all
The following are the main theorems proven in the course. You are expected to be able to give a proof of each of them.
- A finitely generated vector space has a basis and any two bases have the same number of elements.
- Basis = minimal generating set: Extraction of a basis from a generating set;
- Basis = maximal independent set: Completion of an independent set to a basis;
- If U is a subspace of V and dim(U)=dim(V) is finite then U=V.
- If T is a linear mapping from a finite-dimensional vector space U to a finite-dimensional vector space V then
- rank(T)+nullity(T)=dim(U),
- rank(T)+nullity(T^t)=dim(V),
- rank(T^t)+nullity(T)=dim(U).
- The Primary Decomposition Theorem.
- Eigenvalues of an operator = roots of its minimal polynomial;
- Characterization of diagonalizable operators in terms of the minimal polynomial;
- Independence of generalized eigenvectors having distinct eigenvalues.
- The Jordan Canonical Form Theorem. Similarity of real and complex matrices.
- The Spectral Theorem. The diagonalizability of real symmetric and Hermitian matrices.