Linear Algebra in a Nutshell

This is the famous (hehe) lecture from youtube. If you are in my class, you must check this out!

LINEAR ALGEBRA IN A NUTSHELL
                        ((A is square n by n))

Nonsingular                                Singular

A is invertible                            A is not invertible

The columns are independent                The columns are dependent

The rows are independent                   The rows are dependent

The determinant is not zero                The determinant is zero

Ax=0 has one solution x=0                Ax=0 has infinitely many solutions

Ax=b has one solution                    Ax=b has no solution or infinitely many

A has n (nonzero) pivots                 A has r < n pivots

A has full rank r=n                      A has rank r < n

The reduced row echelon form is R=I      R has at least one zero row

The column space is all of R^n           The column space has dimension r < n

The row space is all of R^n              The row space has dimension r < n

All eigenvalues are nonzero              Zero is an eigenvalue of A

A'A is symmetric positive definite       A'A is only positive semidefinite

A has n (positive) singular values       A has r < n singular values

Gilbert Strang