Linear Algebra I-II

I. COURSE DESCRIPTION FROM CATALOG:
A theoretical study of vector spaces, bases and dimension, subspaces, linear
transformations, dual spaces, eigen values and eigenvectors , inner product spaces , spectral
theory, duality, quadratic and bi linear forms . Lec. 3-3. Credit 3-3.

II. PREREQUISITE(S):
MATH 4530 (5530): C or better in MATH 2010 and MATH 3400.
MATH 4540 (5540): C or better in MATH 4530 or 5530.

III. COURSE OBJECTIVE(S):
To introduce students to the theory of linear operators on (mostly) finite dimensional real
and complex vector spaces.

IV. TOPICS TO BE COVERED:

MATH 4530:

0 Systems of Equations and Matrices – Systems of Linear Equations and Matrices – Solution of
Homogeneous Systems – Solution of Inhomogeneous Systems – Matrix Algebra – Systems of Equations
and Matrix Inverses – Fields

1 Vector Spaces – Vector Spaces – Subspaces – Linear Independence – Basis and Dimension –
Coordinates with Respect to a Basis

2 LinearOperators – Preliminary Topics – The Rank and Nullity Theorem – Linear Operators and
Matrices

3 Inner Product Spaces – Preliminaries – Orthogonal Sets – Approximation and Orthogonal Projection –
Applications of Projection Theory – Orthogonal Complements – The Gram Matrix and Orthogonal
Change of Basis

4 Diagonalizable Linear Operators – Eigenvalues and Eigenvectors – Linear Operators with an
Eigenbasis – Functions of Diagonalizable Operators – First- OrderMatrix Differential Equations –
Estimates of Eigenvalues: Gershgorin's Theorems

MATH 4540:

5 The Structure of Normal Operators – Adjoints and Classification of Operators – The Spectral
Theorem – Applications to Matrix Theory – Extremum Principles for Hermitian Operators – The Power
Method – The Rayleigh-Ritz Method – Approximation of a finite number of eigenvalues and eigenvectors
of a Hermitian operator defined on an infinite dimensional space

6 Bilinear and Quadratic Forms – Preliminaries – Classification of Hermitian Quadratic Forms –
Orthogonal Diagonalization – Other Methods of Diagonalization – Simultaneous Diagonalization of
Quadratic Forms

7 Small Oscillations – Differential Equations of Small Oscillations – Undamped Small Oscillations –
Damped Small Oscillations – Galerkin's Method for Partial Differential Equations

8 Factorizations and Canonical Forms – The Singular Value and Polar Decompositions – Applications
of the SVD – Schur's Theorem – Jordan Canonical Form

V. ADDITIONAL INFORMATION:

Graduate credit is earned on the basis of additional work required by the instructor [per
2005-2006 TTU Graduate Bulletin], page 38.

VI. POSSIBLE TEXTS AND REFERENCES:
Schaum’s Outline of Linear Algebra, 4^th edition, Lipschutz
Linear Algebra with Applications , J.T. Scheick
Linear Algebra Done Right, 2^nd edition, by Axler
Advanced Linear Algebra, 2^nd edition, by Roman

VII. ANY TECHNO LOGY THAT MAY BE USED: