Linear Algebra Review

I spent 3 days on reviewing basic linear algebra stuff. I used the online notes from Paul Dawkins and it can be downloaded from http://tutorial.math.lamar.edu/pdf/LinAlg/LinAlg_Complete.pdf.

This book covers most fundamental and important definitions and theorems in linear algebra. The book articulates stuff clearly and precisely. The theorem I like the most is listed below. The proof is elegant (hint: null space).

Suppose that A is an  matrix with linearly independent columns.  Then,  is an invertible matrix.
 
However, some examples are somewhat tedious for me, and the chapter about eigenvalue and eigenvector is kind of short.