_ Linear Algebra: Spring 2007
Chapter 7. Distance and approximation
We have a clear idea on the length of a vector and a distance (or relation) between two vectors in the Euclidean vector space; we want to extend these concepts to the general vector spac
_ Linear Algebra: Spring 2007
Chapter 6. Vector spaces
Generalization of Euclidean vectors; we want to construct an arbitrary vector space, by defining some basic properties to be satisfied.
1. Vector spaces
Definition Let V be a set on which two operati
_ Linear Algebra: Spring 2007
Chapter 5. Orthogonality
Geometric interpretation of matrices and vectors
1. Orthogonality in n-D vector space
Consider a set of standard vectors in a coordinate system, cfw_e1, e2, , en. They possess a couple of good proper
_ Linear Algebra: Spring 2007
Chapter 4. Eigenvalues and eigenvectors
Numerical properties of a matrix
1. Eigenvalues & eigenvectors
Definition Let A be an (n n ) square matrix. A scalar is called an eigenvalue of A if there is a non-zero vector x s.t. A
_ Linear Algebra: Spring 2007
Chapter 3. Matrices
Arithmetic on the matrices based on its algebraic structure Matrix as a transformation (or function)
1. Matrix operations
is the numbers of rows and columns of it and we usually denote it as A (m n ) , wh
_ Linear Algebra: Spring 2007
Chapter 2. Systems of linear equations and matrices
Linear equation
ax + by = c , ax + by + cz = d
Definition A linear equation in the n-variables cfw_x1, x2, , xn is an equation that can be written in the form, a1x1 + a2 x2
_ Linear Algebra: Spring 2007
Chapter 1. Vectors
1. Geometry and algebra of vectors
What is a vector? (1) (Geometric vector) A vector is a directed line segment that corresponds to a displacement from one point A to another point B. We denote it by v = AB