% math 211 - 17/18 - Review/Overview 
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\begin{center} {\Huge\bf Math 211} \end{center} %\vspace{0pt}
\begin{center} {\Large\bf Course Overview} \end{center} %\vspace{0pt}
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\begin{description}   
\item[Chapter 1:] {\bf The Integers}  

-- Divisibility, greatest common divisor

-- The extended Euclidean Algorithm

-- The GCD-criterion and applications, Euclid's Lemma 

-- Application: Solving linear Diophantine equations without/with constraints 

-- Prime numbers and the Unique Factorization Theorem. Applications: 
irrationality, GCD-formula 

\item[Chapter 2:] {\bf Modular Arithmetic} 

-- The calculus of remainders: definition and properties of $a \equiv b $ (mod $m$), the power-mod algorithm, the cancellation law 
 
-- Solving the congruence equation $ax\equiv b\, (\mbox{mod } m)$

-- Modular arithmetic and the ring $\Z/m\Z$; special case: the field $\F_p$

-- The Chinese Remainder Theorem 

-- Fermat's Little Theorem and applications

\item[Chapter 3:] {\bf Polynomials}

-- Complex numbers: complex conjugates, absolute values, geometrical representation,
polar form, multiplication rule, De Moivre's formula, $n$-th roots, Euler's formula  

-- Polynomials: coefficients, degree, the ring $R[x]$, properties of the degree function

-- Divisibility and the division algorithm. Applications: the Remainder Theorem and the Factor Theorem; the substitution method for finding rem$(f,g)$ 

-- The greatest common divisor (of 2 polynomials), the extended Euclidean algorithm; the GCD-criterion, Euclid's Lemma (for polynomials)

-- Irreducible polynomials: basic properties, connection with roots. The Quadratic Formula
(and discriminants)

-- The Unique Factorization Theorem for $F[x]$. Applications: the GCD-formula, the multiplicity of a root   

-- Factoring methods: (a) over $\Q$: the Rational Root Test, Gauss's Lemma, the modular test;
(b) over $\C$: the Fundamental Theorem of Algebra; (c) over $\R$: the Fundamental Theorem of Algebra for $\R[x]$

\item[Chapter 4:] {\bf Interpolation, Approximation and the Geometry of $\R^n$}  

-- The Lagrange Interpolation Polynomial: (i) the formula method; (ii) the linear algebra method; the Remainder Formula 

-- The Least Square Method via orthogonal projection onto a subspace of $\R^n$   

-- The geometry of $\R^n$: (a) points and vectors: dot product, length, distance, angles;
Law of Cosines, Rule of Pythagoras, Cauchy-Schwarz inequality, triangle inequality.
(b) equations of lines, planes, linear sets and of subspaces of $\R^n$. 

-- A distance problem: the orthogonal projection of a vector onto a subspace (matrix method). Application to the Least Square Method

-- Orthogonal and orthonormal bases of a subspace. The orthogonal projection formula 
(via an orthogonal basis). The Gram-Schmidt method for finding an orthogonal basis. 
Orthogonal matrices.


\item[Chapter 5:] {\bf Matrix Polynomials and Discrete Linear Systems} 
\iffalse
-- Evaluating Matrix Polynomials I: (i) Diagonable case: The Diagonalization Theorem: 
eigenvalues, eigenvectors; the characteristic polynomial ch$_A(t)$; determinants.
(ii) Non-diagonable case: Jordan blocks $J(\lm, k)$, Jordan matrices and Jordan's Theorem
\fi

-- Evaluating Matrix Polynomials I: (i) Diagonable case: The Diagonalization Theorem: 
eigenvalues, eigenvectors; the characteristic polynomial ch$_A(t)$; determinants

-- Evaluating Matrix Polynomials I: (ii) Non-diagonable case: Jordan blocks $J(\lm, k)$, Jordan matrices and Jordan's Theorem

-- Evaluating Matrix Polynomials II: The Cayley-Hamilton Theorem. The Generalized Remainder Theorem (substitution method, formula method)

-- Evaluating Matrix Polynomials III: The Spectral Decomposition Theorem. Constituent matrices  
and how to compute them
 
-- Applications: discrete linear systems, difference equations (example: Fibonacci numbers, golden ratio), companion matrix; Markov chains  

\item[Chapter 6:] {\bf The Jordan Canonical Form} 

-- Algebraic and geometric multiplicities of eigenvalues (Sum Formula, Invariance Property) and their relation to the Jordan canonical form $J$ of a matrix $A$

-- Finding $P$ such that $P^{-1}AJ = J$ (for $m\le 3$)

-- Properties of generalized eigenvectors and generalized geometric multiplicities. Application: how to determine $J$ via the method of 2nd differences   
 
\item[Chapter 7:] {\bf Powers of Matrices}  

-- The limit of a sequence of complex numbers; powers of a number 

-- Rules for limits of matrices, power convergent matrices. Eigenvalue criterion for a matrix
to be power convergent

-- Finding $\lim_{n\ra\infty} A^n$. Further properties of constituent matrices 

-- Properties of stochastic matrices and their eigenvalues; primitive stochastic matrices, Perron's theorem, Perron vector; applications to Markov chains.

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