Linear Algebra

These notes represent a summary of some topics in the Anton and Rorres book, Elementary Linear Algebra. David Pierce has begun making these notes in the course of teaching out of this book. Whether he continues them depends on his time and interest, and his ability to format things adequately in HTML.

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 pierced@math.mcmaster.ca.

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 should be italic; letters for vectors should be bold. Mathematical text should be in a fixed-width font. Words being defined should be bold.

I am in the process of figuring out the best way to achieve these ends using style sheets.

On the pages most recently worked on, here is how things appear:

• 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:

  é ê ë

  a11 a12 a13 a21 a22 a23 a31 a32 a33

  ù ú û

  Your browser has to have the symbol font for the brackets on the matrix. I try to make the subscripts the same size as the text on the line, for the sake of horizontal spacing, but Microsoft does not obey the css style sheet properly.
• Instead of using l (the Greek lower-case lambda) for eigenvalues, I'm just using Latin letters like x (since you need the symbol font to see the lambda properly).
• Some words are being defined;
• others are just being emphasized.
• There are some links to other parts of the notes, such as those in the table of contents above.

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