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