Linear Algebra

These notes originated as a summary of some topics in the Anton and Rorres book, Elementary Linear Algebra. I started the notes in the course of teaching out of this book at McMaster University, 1999–2000.

The purpose here is not to reproduce or elaborate on the book or lectures, but to summarize the theory. In particular, examples are generally left to the book and lectures.

Please send corrections to [email protected].

At least one person did send me corrections over the years, when I was at METU. The notes went offline when I moved to Mimar Sinan; now I am reviving them as I prepare to teach linear algebra at the Nesin Mathematics Village.

Contents

• Linear systems
  • The linear in linear algebra
  • Linear systems
• Matrix algebra
  • Linear combination
  • Multiplication
  • Transposition
  • Linear systems reconsidered
  • Inversion
  • Special matrices
• Determinants
  • A definition
  • Properties
  • A technique
  • More properties
  • Another technique
  • Theory
  • A consequence
  • Application to eigenvalues
• Geometry
  • Vectors, points and arrows
  • Norm
  • Dot-product
    • Definition
    • Properties
  • Projections
  • Cross-product
    • Theoretical definition
    • Practical definition
    • Properties
• Higher dimensions
  • n-space
  • Dot-product and norm
  • Dot-product from norm
  • Linear transformations
    • Theoretical definition
    • Practical definition
    • Linear operators
    • Linear transformations as functions
• Abstract vector spaces
  • Formal definition
    • Addition rules
    • Scalar-multiplication rules
    • `Rule of unity'
  • Consequences of the definition
  • Examples
  • Subspaces
  • Linear combinations and spanning sets
  • Linearly independent sets
  • Bases
• Matrix spaces
• Diagonalization
  • Application to differential equations
• Inner-product spaces
• Complex numbers

Typographical conventions

General principles. Letters for scalars are italic; letters for vectors are bold. Mathematical text in general is in a fixed-width font. Words being defined are bold. Here is how things appear in these pages:

• In-line mathematical passages are thus; while
• displayed math passages are this way.
  (They are supposed to be centered.)
• There are scalars such as x, and matrices such as A; and
• there are vectors such as v.
• Theorem. A theorem might appear differently (with a white background);
  (but then again it might not, if I haven't got around to making it that way yet, or if your background color is already white).
• Here is a three-by-three matrix:

  ⌈ | ⌊

  a11 a12 a13 a21 a22 a23 a31 a32 a33

  ⌉ | ⌋

• Some words are being defined;
• others are just being emphasized.

I am not choosing font families or sizes, or the background color of this page; your browser does that. So if things are not quite legible, you can do some adjusting.

First section