Definition:
If A is a nxn matrix, L a scalar is an eigenvalue of A, it there
is a NON-ZERO vector X so that AX = LX. (Usually L is the greek
letter lambda.)

This means 0 is an eigenvalue <=> homogenuous equation AX = 0
has a non-zero solution <=> AX = 0 has oo-many solutions. That
is rref(A) has a column with no pivot.

If we know L is eigenvalue (coming soon) then B = A-LI has
non-zero X so that BX = 0. (because BX = (A-LI)X = AX - LIX
= LX - LX = 0.) So again, finding eigenvectors is the same
as finding non-zero solutions to homogenuous equations with
oo-many solutons.

Definiton:
The vector W is a linear combination of vectors X1, X2, ... Xn
if there are scalars (think reals) C1, C2, ... Cn so that
	W = C1*X1 + C2*X2 + ... + Cn*Xn

This is the same as solving AY = W, where A is the matrix
with A(:,j) = Xj (X1 is the first column of A, X2 the second column ...
Xn the last column of A) and the unknown Y = [C1, C2 ... Cn]^T.
If the system has a solution then W is a linear combination, other
wise it is not a linear combination.

Definition:
The set {X1, X2, ... Xn} is linearly independent (sometimes just
independent) if whenever there are scalars C1, C2, ... Cn
so that C1*X1 + C2*X2 + ... Cn*Xn = 0, then C1 = C2 = ... = Cn = 0.

This is the same as solving AY = 0, where A is the matrix
with A(:,j) = Xj (X1 is the first column of A, X2 the second column ...
Xn the last column of A) and the unknown Y = [C1, C2 ... Cn]^T.
If the system has a unique solution (which must be C1=...=Cn = 0)
then the set is independent, other it is not independent.

Definition:
A set which is NOT independent is said to be dependent



We used this to obtain two independent solutions of AX=0 when
rref(A) had two columns without pivots.


We did deep background showing for every lambda, d/dx has lambda
as an eigenvalue with eigenfuction exp(lambda x). The equation
AX = lambda X became the ODE y' = lambda y in this case. 

We also say the sin(x) and cos(x) solutions of (D^2+I)y = 0
had stuff in common with finding two independ solutions of AX = 0.