Math508 Summer 2003 (05/27/03 - 08/01/03) Applied Linear Algebra

Course Information:

Instructor:	Shangbing Ai, MDH 320, Phone: 824-2229, Email: ais@email.uah.edu
Class meetings:	MW 5:00-7:00 pm, MDH 300
Office Hours:	MW 1:30-4:30 pm or by appointment
Prerequities:	MA 244, MA 324. No credit given to students who have successfully completed MA 544.
Textbook:	Matrix Theory and Applications with MATLAB, by Darald J. Hartfiel.
Grading:	Quizzes and Matlab homework (15%), two one-hour tests (each 25%), the final (35%).
Final Test:	The final is comprehensive and will be given on July 30, 2003 from 6:30 -- 9:00 pm.

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Course Objectives:

Solving systems of equations, finding inverses of matrices, determinants, vector spaces, linear transformations, eigenvalues and eigenvectors, normal matrices, canonical forms of matrices, applications to systems of linear differential equations, and use of computer software such as MATLAB.


Main Topics and Schedule (tentative)

Most of materials in the first three chapters of the book has been covered in Math 244. So we will spend as less time as possible on those three chapters. It is a good idea for you to refer to the text book for Math 244 while we study those chapters.

Chapter 1. Review of Matrix Algebra (5/28)

• Matrix Algebra (algebra for complex numbers, Euler's formula, matrix with complex entries, block matrix, matirix multiplication. See examples 1.2, 1.3)
  
• System of Linear Equations (pivot entries, row echelon form matrix, row operations for matrices, Gaussian elimination, solving linear systems, finding the inverse of a matrix. See examples 1.4, 1.5)
  
• Determinant (cofactors, adjoint matrix, derterminant by row expansion or column expansion, example 1.7)
  
• MATLAB (solving Ax=b) (matlab command: A\b)
  
• Exercises (pp. 22-28): 1, 6 (a), 9, 14, 18, 30 -- 33. Matlab problems 30-33 will be collected on June 4.

Chapter 2. Introduction to Vector Spaces (6/2 -- 6/4)

• Vector Spaces (definitions for vector spaces, subspaces, the span S, exampes 2.3-2.6, 2.8)
  
• Exercises (pp.40-43): 6, 7, 12, 13.
  
• Dimension (linearly independent and linearly dependent, a basis, examples 2.92.14)
  
• Exercises (pp. 55-58): 1, 3, 10, 11, 14, 15, 19, 22
  
• Linear Transformations(Definition for linear transformations, one-to-one linear transformation, the null space, Theorems 2.7, 2.8, examples 2.18-2.20)
  
• Exercises (pp. 70-73): 2, 4, 6, 8, 12, 15
• Quiz 1 ( quiz 1 is given on Wed., June 4. The problems are to sove a linear system by the Gaussian eliminations and find the determinant of a matrix)
• Solution to Quiz 1
• Quiz 2 (Do the problem 22 by hand on Page 57-58 with all possible details. For (d), find also linear combinations in terms of the basis you found for those given matrices not in the basis. Please turn in your work in the class on Wed. June 11.)
• Solution to Quiz 2

Chapter 3. Similarity

• Nonsigular matrices (nonsigular matrix, rank, vandermonde matrix and its determinant, Theorem 3.2, Corollaries 3.2, 3.3, Examples 3.1-3.3.)
• Exercises (pp.85-87): 3, 4, 5, 14
• Diagonalization (chracteristic polynomial, eigenvalues, eigenvectors, algebraic and geometric multiplicities of eigenvalues,eigenspaces, diagonalization)
• Exercises (pp.97-99): 1, 4, 7, 8, 9, 12, 13
• Conditions for diagonalization (left eigenvectors)
• Exercises (pp. 107-108): 1, 4, 5, 6, 7, 9, 11, 12
• Jordan Forms
• Exercises (pp.114-115): 1, 2, 3, 4, 5, 6, 8, 9, 10, 13
• Quiz 3 on 6/18 (diagonalization of a 3 x 3 matrix, and recognition of Jordan matrix)

Chapter 4. Matrix Calculus

• Calculus Matrices (convergence of a sequence of Matrices, convergence of a matrix series, Theorem 4.4)
• Exercises (pp. 126-128): 2, 5, 6, 10
• Difference Equations
• Exercises (pp. 139-142): 1, 2, 3, 8, 9
• Hand in homework on 6/23: 3, 8, 9 on Pages 139-141.
• Differential Equations (Lemma 4.1, Example 4.10, exp(Jt), Examples 4.11-4.12)
• Exercises (pp: 151-156): 1, 5, 7, 8, 10, 16
• Test 1 on 6/30 (I have removed some problems from each of the exercises. You are supposed to be able to do all the remaining problems. The problems in Test one will be similar to those problems and the quiz problems we had. We will go over some of them in class on 6/25.)
• Solution to Midterm Test

Chapter 5. Normed Vector Spaces

• Vector Norms (vector norms, distance (or metric) induced by a vector norm between two vectors, p-norms (p>=1) and infinite norm)
• Exercises (pp. 164--166): 1-5, 9, 10, 12, 13, 14
• Induced Matrix Norms (Induced matrix norm (Def. 5.2), properites of induced matrix norms (Thm. 5.4), formulas for induced matrix norms by 1-norm, 2-norm, infinity norm (Thm. 5.5), the condition number of a matrix)
• Exercises (pp. 176-179): 4, 5, 6, 7, 8, 10, 12, 13, 15, 16, 17
• Some special Norms (Frobenius norm, R-norm for matrices and vectors, spectral estimates by matrix norms, Theorem 5.9, Corollaries 5.1, 5.2)
• Exercises (pp. 188-190): 1, 3, 4, 5, 6, 7, 9
• Inner Product and Orthogonality (definition of the inner product, orthogonality of a set of vectors, independency of a set of nonzero vectors, Gram-Schmidt orthogonality process, Fourier coefficients and sum, QR-factorization of matrix, orthogonal projection of a vector to a subspace, othogonal projection matrix)
• Exercises (pp. 201-203): 1, 3, 7, 8, 9, 10, 12, 13, 14, 17, 19
• Quiz 5 on 7/14 (The quiz problems will be based on the problems assigned in Sections 5.1, 5.2, 5.3)
• Solution to Quiz 5
• Hand-in homework on 7/21: 13, 14, 22 on Page 202-203. (Note: No MatLab is needed for all three problems.)

Chapter 6. Unitary Similarity

• Unitary Matrices (Unitary matrix, orthogonal matrix, rotation matrix, reflection matrix, Householder matrix, properties of unitary matrices (Theroems 6.1-6.5)
• Exercises (pp.215-218): 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15
• Schur Decompositions (extending an orthogonal set to an orthogonal basis (Lemma 6.2), Schur's decomposition Theorem (Thm 6.6), Corollaries6.1-6.2, Lemmas 6.3-6.4), Corollary 6.3)
• Exercises (pp. 232-234): 1, 2, 5, 6, 7, 8, 9, 12, 13
• Hand-in homework on 7/23: 1 (b), (c) on Page 232. (Solution to 1 (c))
• Hand-in homework on 7/28: 5 (a), (b) on Page 232. (Solution to 5 (a))
• The final on 7/30 (Gauss-Jordan elinimination, the row reduced echelon matrix, a basis for the solution space of Ax=0, use the Gauss-Jordan elinimination to find the inverses of nonsingular matrices, determinant, conjugate transpose of a matrix, the absolute value of a complex number; Vector space, subspace, linearly independent and dependent, bases and dimensions of vector spaces and subspaces, methods for finding bases, column vector space of a matrix, linear transformations, null space N(A) and range R(A) of a matrix A; the rank of a matrix, extending a set of vectors to a basis for E^n, the Vandermonde matrix and its determinant, eigenvalues, eigenvectors, eigen-space, algebraic and geometric multiplicity of an eigenvalue, similarity, diagonlization (by the eigenvalue-eigenvector method), Jordan canonical matrix, Jordan blocks; Computations of A^k and e^A; definition of norms on vector spaces, 1-norm, 2-norm, infinty-norm for vectors in E^n, and corresponding induced matrix norms, inner product in E^n, orthogonality, orthogonal projection of a vectorto a subspace, the closest vector in a subspace to a given vector, projection matrix in E^n, Gram-Schmidt process, finding an othogonal basis for a subspace W=span{a_1,a_2,...,a_m}; Unitary matrix, rotation matrix, reflection matrix, the Housholder matrix, properties of unitary matrices, unitary similarity, extension of an orthogonal set to a basis for E^n, the Schur decomposition, normal matrix, properties of normal matrices, perperties of Hermitian matrices, eigenvalue-eigenvectors method to unitarily diagonalize a normal matrix)

Chapter 7. Singularly Valued Decomposition

Chapter 8. LU and QR Decompositions

Chapter 9. Properties of Eigenvalues and Eigenvectors

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References:

Bernard Kolman, Introductory Linear Algebra with Applications, Sixth Edition. (This is the text book for Math 244.)