Summary of textbook sections covered.

E stands for Edwards & Penney. L stands for Leon.

Wed Sept 1: E1.1
Fri Sept 3: E1.3
Comments for Week 1:
Some terminology learned:
order of a differential equation
n-parameter family of solutions
linear and nonlinear equations
slope fields
Slope field example handed out in class on Friday.
Wed Sept 8: E1.4
Fri Sept 10: E1.5
Comments for Week 2:
Learned how to solve separable equations
(which are a particular type of first-order equation).
Learned how to find an integrating factor
(for a 1st order linear eqn with non-constant coeffs).
Mon Sept 13: E2.1
Wed Sept 15: E3.1
Fri Sept 17: E3.2 / E3.3
Comments for Week 3:
Some population models, including the logistic equation
The logistic equation can be solved explicitly by hand, though it's cumbersome.
Brief mention of stable and unstable equilibrium points
Handout illustrating stable and unstable equilibria
The characteristic equation of a lin. homog. diff eqn with constant coeffs
Using the char. eqn. to find general solution of the diff. eqn., including:
what to do if there are repeated roots or complex roots
A brief introduction to differential operators
Mon Sept 20: E3.2 / E3.3
Wed Sept 22: E3.4
Fri Sept 24: E3.5
Comments for Week 4:
There's an existence-uniqueness theorem for higher order linear diff. eqns.
The Wronskian is a determinant that tests if functions are linearly dependent.
Defintion of the complementary function of a differential equation
Motion of a mass attached to a spring and dashpot:
undamped versus damped motion, and free versus forced
Using undetermined coefficients to find particular solns to nonhomog. diff. eqns.
Mon Sept 27: E3.7
Wed Sept 29: L1.1
Fri Oct 1: TEST 1
Comments for Week 5:
Simple RLC series circuit and corresponding differential equation:
RLC equation is mathematically the same as for mass-spring-dashpot.
Solution is sum of transient current and steady periodic current.
Systems of linear equations, coefficient matrix, augmented matrix
Elementary row operations, pivot, strict triangular form
Mon Oct 4: L1.2
Wed Oct 6: L1.3
Fri Oct 8: L1.4
Comments for Week 6:
Row echelon form, homogeneous and nonhomogeneous linear systems
Leading 1's allow us to define leading variables and free variables.
REDUCED row echelon form (where you also have zeros ABOVE leading ones)
Overdetermined and underdetermined systems (how many solutions can they have?)
Example of application of linear algebra to electric circuits
Definition of Rn, matrix addition, scalar multiplication, matrix multiplication
Powers of a square matrix, n by n identity matrix, transpose of a matrix
Inverse of a square matrix -- A given matrix may or may not have an inverse.
A square matrix can be nonsingular (invertible) or singular (noninvertible)
Mon Oct 11: L2.1
Wed Oct 13: L2.2
Fri Oct 15: L3.1
Comments for Week 7:
Determinant of a square matrix. 2x2 and 3x3 have nice formulas, larger ones don't.
Minors and cofactors. Minor or cofactor expansion along a row or column.
Rules for how the three types of row operations affect the determinant of a matrix.
Rules for the determinant of the transpose or the inverse of a matrix.
Rule for the determinant of a triangular matrix.
A vector space must be closed under vector addition and scalar multiplication.
Vectors can be n-tuples in Rn, functions, or other things.
Some subsets of a vector space can also be a vector space.
Mon Oct 18: L3.2
Wed Oct 20: L3.3
Fri Oct 22: L3.4
Comments for Week 8:
Some subsets of a vector space can also be a vector space -- i.e., a subspace.
A subset is a subspace if it is closed under addition and scalar multiplication.
The solutions of a linear homogeneous differential equation are a subspace
of the space of all continuous functions -- even if we can't solve the equation!
Definition of the null space of a matrix.
An m by n matrix can define a function from Rn to Rm.
The null space of an m by n matrix is a subspace of Rn.
Definition of the span of a set of vectors.
Definition of a linearly independent set of vectors.
Definition of a basis of a vector space.
In Rn, what can and can't happen if you have a set containing:
fewer than n vectors, more than n vectors, or exactly n vectors?
Mon Oct 25: TEST 2
Wed Oct 27: L3.5
Fri Oct 29: L3.6
Comments for Week 9:
There are many possible bases for Rn (or for any n-dimensional space).
Given a basis for Rn, any vector in Rn has coordinates with respect to that basis.
A transition matrix can be used to convert between two different bases.
For any matrix, we can define its row space and its column space.
The row space of an m by n matrix is a subspace of Rn.
The column space of an m by n matrix is a subspace of Rm.
To find a basis for the row space or the column space, do row operations.
(For a basis for the row space, use appropriate rows of the reduced matrix.)
(For a basis for the column space, use appropriate rows of the original matrix.)
Mon Nov 1: L6.1
Wed Nov 3: L6.3
Fri Nov 5: L6.3
Comments for Week 10:
Definition of eigenvalue and eigenvector.
Eigenvalues and eigenvectors are defined only for square matrices.
Eigenvectors, by definition, must be nonzero vectors.
The eigenvalues of a matrix A are the numbers λ
that make the matrix (A−λI) have determinant zero.
We refer to det(A−λI) as the characteristic polynomial.
The algebraic multiplicity of an eigenvalue is the number
of times it appears as a root of the characteristic polynomial.
The geometric multiplicity of an eigenvalue is the number
of independent eigenvectors belonging to that eigenvalue,
or the dimension of the corresponding eigenspace.
We can diagonalize a square matrix if every eigenvalue
has geometric multiplicity equal to its algebraic multiplicity.
(Lines 4, 5, and 6 contain the letter lambda -- hope it works in all browsers)
Mon Nov 8: L6.3
Wed Nov 10: E4.1
Fri Nov 12: E5.1
Comments for Week 11:
Diagonalization, when possible, helps us compute powers of a matrix
as well as the exponential of a matrix.
We can find both specific powers and a general formula for the kth power.
We skipped most of 6.4. We just singled out a few facts about eigenvalues.
SYSTEMS of first-order differential equations:
There are standard tricks for rewriting a high-order differential equation
as a system of first-order differential equations.
The system can be written in matrix form, and may be either
homogeneous or nonhomogeneous.
For a homogeneous system, there's a technique for finding the general solution
using eigenvalues and eigenvectors.
Mon Nov 15: E5.2
Wed Nov 17: TEST 3
Fri Nov 19: E5.4
Comments for Week 12:
Initial conditions for a system of diff eqns are most often at t=0,
which makes the math fairly easy, but you still do a bit of linear algebra.
Complex eigenvalues are a little tricky:
-- eigenvector will contain complex numbers
-- one complex eigenvalue will lead to two real solutions
-- use Euler's formula to split complex solution into real and imaginary parts.
Repeated eigenvalues are a little tricky:
-- form of solution is a bit more complicated
-- solution contains t times eigenvector plus generalized eigenvector
Mon Nov 22: E5.4
Comments for Week 13: see above
Mon Nov 29: E5.5
Wed Dec 1: E6.1
Fri Dec 3: E6.1
Comments for Week 14:
Definition of a fundamental matrix Phi(t) for a system of diff eqns.
The product Phi(t) times the inverse of Phi(0) is useful for initial value problems.
There is a relationship between that product and matrix exponentials.
Definition of an autonomous system of diff eqns.
Given a second-order autonomous system with x = x(t), y = y(t),
we can form a slope field, which might tell us something
about the solutions even if we can't solve the system explicitly.
Definition of a critical point or equilibrium point of a
(not necessarily linear) system of diff eqs. Finding critical points is
"just algebra", and typically there are a small finite number of them.
One can try to determine the behavior of a system near a critical point
by looking at the slope field, or by looking at the linear approximation
of the system near the critical point.
In fact, the eigenvalues of the linear approximation can help you classify a
critical point as a sink, source, saddle point, etc., but we didn't get into the details.
Mon Dec 6: REVIEW
Wed Dec 8: REVIEW
Comments for Week 15: see above