Math 641-600 Midterm Review
The midterm will be given on October 14 and will consist of an in-class
part and a take-home part. The test will cover sections 1.1-1.4, 2.1,
2.2.1-2.2.2. It will also cover the material done in class on the Lebesgue
integral.
The in-class test will consist of the following: statements of theorems
and definitions; short problems or propositions similar to homework
problems or examples done in class; and either a critical part or
sketch of a proof for one of the major theorems proved. The take-home test
will have longer computations, proofs, or problems.
Linear algebra
- Inner product spaces & normed linear spaces
- Subspaces, orthogonal complements
- Orthogonal sets of vectors, the Gram-Schmidt procedure
- Least squares, minimization problems, projections, normal equations
- Self-adjoint matrices & their properties
- Estimation of eigenvalues
- Rayleigh-Ritz maximum principle
- The Courant-Fischer Minimax Theorem and applications
- The Fredholm Alternative
Banach & Hilbert spaces
- Convergent sequence, Cauchy sequance, complete spaces - Hilbert spaces and Banach spaces
- Lebesgue measure and integral, measurable functions, the Monotone and Dominated Convergence Theorems
- Special (complete) spaces - lp, Lp (1 ≤ p ≤ ∞), C[a,b],
Ck[a,b], Sobolev space Hn[a,b], Sobolev-type inequalities.
- C[a,b] and properties
- Modulus of continuity
- Density of splines
- Weierstrass Approximation Theorem — density of polynomials in C[a,b], Bernstein polynomials
- Density of polynomials in Lp, 1 ≤ p < ∞
- Hilbert spaces
- Minimization problems, least squares - discrete and continuous -, variational/finite element methods, normal equations
- Complete sets of orthogonal functions, Parseval's identity
- Orthogonal polynomials & completeness
Updated 10/9/09 (fjn).