Checklist of things you should learn by the end of Math 2250
This is a good starting point, not necessarily exhaustive (but close), for assessing your
knowledge in this course.
1. If x is a real dependent variable, and t is the independent variable, be able to explain
to your 75-year-old grandmother what
“
dx
= f (x)”
dt
means in plain language (assume for the sake of argument has no advanced math background but is wise beyond her many years).
2. The definitions and be able to distinguish three categories of first-order one-dimensional
DEs: directly integrable, separable, and first-order linear. Know the techniques to solve
each types.
3. Know the definitions of general and particular solutions for DEs in the context of
first-order DEs and contrast these definitions with the same terms in the context of
higher-order homogeneous and non homogeneous DEs.
4. Be able to explain why performing a row operation on a linear system allows one to
solve for the unknown vector of values.
5. Know the definition of a linear operator in the context of euclidean vector spaces and
function spaces. Give examples of each.
6. Know the definition of a homogeneous and non-homogenous linear equation for both
vectors and functions and give examples of each.
7. What are the essential elements/properties that define a vector space V ?
8. What conditions/tests must be satisfied for a set S of vectors in a vector space V to
be a subspace? Give examples of sets that are and are not.
9. What is the smallest possible subspace you can come up with for a vectors space V ?
10. Know how to use the term “span” as a verb and a noun. When do you use it as a
noun, how do you use it as a verb?
11. Suppose we create the set of all linear combinations of two vectors in 3D. Can you
describe what that set is in plain language? How can the word “span” be used in this
context?
12. Can the word “basis” be used as a verb, or a noun? What can a basis do?
13. What three solution scenarios are possible for a general linear matrix equation Ax =
b? What types of solution scenarios are not possible? If a matrix is invertible, what
restricted set of scenarios are possible? What if b = 0?
14. Suppose you perform row operations on a matrix and after the matrix is fully reduced
to echelon form, there is a row of zeros. What type of subspaces are possible from the
equation Ax = 0?
15. Know how to find a basis for a solution subspace defined as all vectors such that
Ax = 0.
16. What information can be gained from the determinant of a matrix both in geometric
and in algebraic contexts?
17. In which situations can the determinant be used to determine if a set of vectors is
linearly independent or not?
18. Know the procedure for finding particular solutions to non-homogeneous operators
Lx = f , when f is a exponential function, trigonometric, or a polynomial, or a product
of all three.
19. Suppose Lx = f is solved by x1 and also x2 . Under what conditions of f does x1 + x2
also solve the equation?
20. Suppose Lx = f , where f 6= 0 is solved by x1 and that x1 + x2 also solves the same
equation. What kind of solution must x2 be?
21. Know the definition of linearity and the superposition of solutions. What kinds of
linear equations admit linear superpositions of solutions?
22. The Laplace transform is a linear operator. Know the explicit definition of the transform in terms of integration.
23. Know how to compute the energy spectral density of a function f (t) by using the
Laplace transform.
24. Know the time and s-shift identities for the Laplace transform.
25. Know the convolution theorem for the Laplace transform and how to use it to solve
ODEs.
26. Know how to transform a linear ODE IVP of x(t) into an algebraic equation involving
an unknown function X(s) and how to transform back.
27. Know the Laplace transform for special functions such as tn , exponentials, trigonometrics, the step function u(t − a) and the delta functional δ(t − a).
28. What kinds of operations can be performed on the equation A − λI, for λ ∈ C? When
is it invertible? When is it non-invertible? How can you tell?
29. What conditions on eigenvalues λj for j = 1, ..., n guarantee a linearly independent
eigenvector basis for Rn ?
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30. What does it mean for A if one of the eigenvalues λ equals zero? What if none of the
eigenvalues equal zero?
31. Suppose A = AT . What can be said about the eigenspace, and values?
32. Know the definition of matrix similarity A ∼ B.
33. Know how to diagonalize a matrix A, if possible. That is, how do you define D, P
and P −1 ? If A is diagonalizable, what is it similar to?
34. Given a diagram and parameters indicated, be able to write down an appropriate firstor second-order DE that describes the state of the system for, respectively, mixture-tank
problems and mass-spring systems.
35. Given a mass-spring-dashpot system, be able to write down a first-order DE system
that describes the position and velocity states of the system.
36. What do eigenvalues describe for first-order and second-order linear DE systems?
37. Be able to use eigenvector-value analysis to write down general solutions to first- and
second-order DEs.
38. Given a forcing function F~ = f~ cos(ωt), be able to find the particular solution to the
equation ~x00 = A~x + F~ . What does this reduce to when ω = 0? What if ω > 0?
39. What is a Jacobian matrix and where does it come from?
40. Given a non-linear system ~x0 = f~(~x), what is an equilibrium point ~x∗ and how do you
determine its stability?
41. Be able to categorize a equilibrium point in 2D as stable, unstable; and saddle points,
spirals, and non-spiral points. Be able to draw pictures of each.
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