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
