Linear Algebra Review
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Lecture 1, Sep. 5, 2007
Basics of Linear Algebra:
vector spaces, linear combinations, linear dependence and independence, bases
and coordinates.
Linear transformations and matrices, matrix multiplication,
coefficient matrices and systems of linear equations, ker
and Im of a linear transformation, change of basis
and coordinate transformation matrices.
Lecture 2,
Sep. 7, 2007
Inner
products, norms, the dual space, the Riesz
representation theorem. Cauchy-Schwartz inequality, triangle inequality. Orthonormal bases, orthogonal complements and orthogonal projections.
The Gramm-Schmidt algorithm.
Orthogonal transformations and orthogonal matrices.
Eigenvectors
and Eigenvalues, determinants and characteristic
polynomial. The
Spectral Theorem.
Lecture 3,
Sep. 10, 2007
Matrix decompositions: the
singular value decomposition (SVD), the QR-decomposition, orthogonal
projections and Householder reflections
Lecture 4,
Sep. 11, 2007
Numerical experiments with
MATLAB.
Least squares problems.
Numerical Analysis: floating point numbers, algorithms, stability and backward
stability. Condition numbers, well and ill-conditioned problems,
ill-conditioning of root finding. Condition number of a
matrix. LU-decomposition without and with pivoting.
Lecture 5,
Sep. 12, 2007
Cholesky decomposition.
Algorithms for finding eigenvalues
and eigenvectors. Rayleigh quotient, power iteration, inverse iteration, Rayleigh
inverse iteration.
Lecture 6,
Sep. 14, 2007
Regression Analysis
Linear Models, Ordinary Least Square Estimators,
Statistical Properties of OLS Estimators
Lecture 7,
Sep. 17, 2007
Efficient and Unbiased Estimators
The Gauss-Markov Theorem, Hypothesis Testing in Linear Models