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 david.pierce@msgsu.edu.tr.
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
 Dotproduct
 Definition
 Properties
 Projections
 Crossproduct
 Theoretical definition
 Practical definition
 Properties
 Higher dimensions
 nspace
 Dotproduct and norm
 Dotproduct from norm
 Linear transformations
 Theoretical definition
 Practical definition
 Linear operators
 Linear transformations as functions
 Abstract vector spaces
 Formal definition
 Addition rules
 Scalarmultiplication rules
 `Rule of unity'
 Consequences of the definition
 Examples
 Subspaces
 Linear combinations and spanning sets
 Linearly independent sets
 Bases
 Formal definition
 Matrix spaces
 Diagonalization
 Application to differential equations
 Innerproduct spaces
 Complex numbers
Typographical conventions
General principles. Letters for scalars are italic; letters for vectors are bold. Mathematical text in general is in a fixedwidth font. Words being defined are bold. Here is how things appear in these pages:
 Inline 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 threebythree matrix:
⌈

⌊a_{11} a_{12} a_{13}
a_{21} a_{22} a_{23}
a_{31} a_{32} a_{33}⌉

⌋  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.