Linear algebra and vector Calculus

Lecture Note

What's New on this Course ?

 

This course helps students to master the material of a undergraduate-level linear algebra and vector calculus. The lecture will coverer Gaussian reduction, vector spaces, linear maps, determinants, eigenvalues and eigenvectors, and vector calculus. The audience must have a background of at least one semester of Calculus.


Course History:

(1) January 20, 2007-June 20, 2007

(2) January 20, 2008-June 20, 2008


Lecture notes have been posted whenever possible.

Grading:4 homeworks (60%) and final examination (40%). Students must fluently grasp the use of any of the following mathematical packages: Mathematica, or Maple, or Matlab.

 

LEC #

TOPICS

1

Chapter One: Linear Systems ( Lecture Note: pdf - 599 kB)

I Solving Linear Systems : 1. Gauss’ Method

2. Describing the Solution Set

3. General = Particular + Homogeneous

2

II Linear Geometry of n-Space: 1. Vectors in Space; 2. Length and Angle Measures

III Reduced Echelon Form: 1. Gauss-Jordan Reduction; 2. Row Equivalence

(pdf Homework Set 1---27 kB, Due: March 21, 2008)

3

Chapter Two: Matrices and Determinants (Lecture Note: ppt - 1277 kB)

I. Definition, and Operations of Matrices

1. Sums and Scalar Products; 2. Matrix Multiplication; 3. Mechanics of Matrix Multiplication; 4. Inverses

4

II Definition of a Determinant: 1. Exploration; 2. Properties of Determinants;

III Geometry of Determinants: Determinants as Size Functions

(Lecture Note: ppt - 1136 kB)

(pdf Homework Set 2 --- 34kB, Due: May 9, 2008 )

5

Chapter Three: Vector Spaces (Lecture Note: PPT - 1277kB)

I Vector Spaces and Subspaces

II Spanning Sets and Linear Independence

6

III Basis and Dimension

IV Rank of a Matrix

V Change of Basis: Changing Representations of Vectors

7

VI Inner Product Space: Inner Product, Length, Distance and Angles (Lecture Note: PPT - 1657 kB)

VII Orthonormal Basis: Gram-Schmidt Process

VIII Mathematical Models and Least Squares Analysis

8

Chapter Four: Linear Transformations (Lecture Note: PPT - 3752 kB)

I Introduction to Linear Transformation

II The Kernel and Range of a Linear Transformation

III Matrices of Linear Transformations

IV. Transition Matrices and Similarity

(pdf Homework Set 3 --- 36 kB, Due: May 28, 2008)

9

Chapter Five: Eigenvalues and Eigenvectors (Lecture Note: PPT - 679 kB)

I Eigenvalues and Eigenvectors

10

II Diagonolization

III Symmetric Matrices and Orthogonal Diagonalization

11

Chapter Six: Singular Value Decomposition (SVD) (Lecture Note: doc - 4806 kB)

I Usage of SVD;

II Basic Idea of SVD;

12

III An Example of SVD

IV Some Applications of SVD

13

Chapter Seven: Vector Calculus (Lecture Note: PPT - 972 kB)

I Fluid Flow: 1.Geneal FLow and Curved Surfaces

II Vector Derivatives: Div, Curl, and Strain

III Computing the Divergence and the Curl of a Vector Field (Mathematica Notebook: vf_div_curl.nb - 607 kB)

14

IV Gradient

V Integrals: Line Integral and Surface Integral (Line Integral: nb - 647 kB) (Surface Integral : nb - 976 kB)

VI Gauss's Theorem and Stokes' Theorem (Divergence Thm: nb - 398 kB) (Stokes Thm : nb - 90 kB)

VII Application Examples of Vector Calculus

(pdf Homework Set 4 ---Due: June 20, 2008)

( Final Examination --- 2007)

( Final Examination --- June 20, 2008)