Rank Theorem Examples, Discrete Linear Dynamical System Example (Eigenvalues and Eigenvectors)

(a.k.a. Differential Equations with Linear Algebra, Lecture 21B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 21B).

#ranktheorem #dynamicalsystems #linearalgebra

Similar problem statements from David Lay's Linear Algebra textbook:

Suppose the solutions of a homogeneous system of five linear equations and six unknowns are all multiples of one nonzero solution. Will the the system necessarily have a solution for every possible choice of constants on the right sides of the equations?

Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations' right sides to make the new system inconsistent?

Suppose a nonhomogeneous system of nine linear equations and ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other?

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution?

A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations?

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side?

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will still have a solution?

(0:00) Lecture overview
(0:59) Definition of the rank of a matrix A
(1:30) Rank Theorem statement (a.k.a. Rank-Nullity Theorem)
(3:00) Applications of the Rank Theorem
(12:00) A linear system of difference equations
(15:25) But how do we compute A^n?
(19:39) Guess solutions of the difference equation
(24:51) Key eigenvalue/eigenvector equation.
(28:54) Two linearly independent solutions
(34:00) General solution is obtained as a linear combination of the two (by the Basis Theorem)
(34:49) Solve a generic initial-value problem (IVP)
(37:55) Use this to find A^n (the nth power of the square matrix A)

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[georgesadler]

Professor Kinney, thank you for a solid example of the Rank Theorem and the Discrete Linear Dynamical System Examples. I also encountered Difference Equations in Signal and Systems Theory and Control Systems. Difference Equations and the use of Linear Algebra are used to describe many systems in Science and Applied Engineering.