## Plan

## 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