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Eigenvalues and Eigenvectors
(See MATH 323 Lecture 24#Eigenvalues and Eigenvectors→)
Just to mention it here,
Eigenvalues
are solutions to 
Eigenvectors are nonzero solutions
to
Exercise 5
Find eigenvalues and eigenvectors of
Find eigenvalues and eigenvectors of
Solving Systems of Equations
Theorem 7.4.1
Let
be
solutions to the homogeneous system
on the interval
, where
is a continuous
matrix function on
,
then for any real numbers
, the vector
is a solution to the homogeneous system.
Theorem 7.4.2
Let
be
linearly independent solutions to the homogeneous system
on the interval
, where
is a continuous
matrix function on
,
Any solution
can be expressed in the form
where
are constants.
Theorem 7.4.3
The Wronskian
is either 0 on
or never vanishes on
.
In other words, if
, then the solutions are linearly dependent and do not describe all possible solutions.
Otherwise, we have found all solutions