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.
MPSetEqnAttrs('eq0063','',3,[[23,6,0,-1,-1],[30,8,0,-1,-1],[39,10,0,-1,-1],[35,9,0,-1,-1],[46,12,0,-1,-1],[57,14,0,-2,-2],[97,24,0,-3,-3]]) MPEquation() MPSetEqnAttrs('eq0064','',3,[[21,10,0,-1,-1],[27,13,0,-1,-1],[34,18,0,-1,-1],[31,14,0,-1,-1],[40,20,0,-1,-1],[52,23,0,-2,-2],[84,40,0,-3,-3]]) MPEquation() MPSetEqnAttrs('eq0064','',3,[[21,10,0,-1,-1],[27,13,0,-1,-1],[34,18,0,-1,-1],[31,14,0,-1,-1],[40,20,0,-1,-1],[52,23,0,-2,-2],[84,40,0,-3,-3]]); MPDeleteCode('eq0064') MPSetEqnAttrs('eq0063','',3,[[23,6,0,-1,-1],[30,8,0,-1,-1],[39,10,0,-1,-1],[35,9,0,-1,-1],[46,12,0,-1,-1],[57,14,0,-2,-2],[97,24,0,-3,-3]]); MPDeleteCode('eq0063')
However, some examples are somewhat tedious for me, and the chapter about eigenvalue and eigenvector is kind of short.