------Plan
Syllabus
1.2
1.3
Introduction
This class is about linear transformations.
The goal of this course is to provide a deeper understanding on linear algebra.
Application and computations.
Course Outline
Chapter 1 Vector spaces
Chapter 2 Linear transformations
Chapter 3 Matrix operations
Midterm around here or before this?
Chapter 4 Determinants
Chapter 5 Diagoanlization
Chapter 6 Inner products
1.2 Vector spaces
Informally, a scalar is a quantity represented by a single number. For example, mass, speed, length.
The scalars live in a field. This is not important. See appendix C. It is a number system where you can add, multiply, subtract, and divide.
R, C primary examples.
Q, finite fields
not Z or N
Informally, a vector is a list of scalars that you can scalar multiply and add together. And a vector space is a set of vectors with some extra properties.
R^n, functions
can represent a few things, (age, weight, height)
Defintion: A vector space is a set of elements (called vectors) with additional and scalar multiplication defined with these additional properties:
(VS 1) Addition commutes
(VS 2) Associativity of Addition
(VS 3) Existence of 0
(VS 4) Existence of inverses
(VS 5) 1 acts identically
(VS 6) (ab)x=a(bx)
(VS 7) a(x+y)=ax+ay dis p.
(VS 8) (a+b)x dis p.
Examples
F^n for any field F
all functions
continuous functions on R
set of matrices
polynomials of bounded degree
polynomials of any degree
nonexample, (x1, x2)+(y1, y2)=(x1-y1, x2+y2)
nonexample, a circle
C is a 2-dimenisional real space
We use 0 to denote many things in this class.
We can write u+v+w without worry.
Theorem: If x+y=x+z then y=z.
Corollary: 0 and inverses are unique. make a minus sign comment here.
Theorem: 0x=0, a0=0, (-a)x=-(ax)=a(-x), here -1 is addivitive inverse
1.3 Subspaces
A subspace is a subset that is also a subspace.
The whole space and the zero subspace are always subspaces
A subset is a subspace if and only if these 3 conditions hold
Examples
continous functions > diff > smooth > polynomials
symmetric matrices
nonexample, invertible matrices
come up with your own?
Intersections