The final exam be on Tuesday, January 14th at 9:15am in Harvard Hall room 104

Problem review sessions (to go over practice exam problems, and answer any general questions you might have:
Monday, Jan. 6th from 4 to 5:30 in Hall E
Friday, Jan. 10th, from 5 to 6:30 in Hall D
Saturday, Jan. 11th from 2 to 3:30 in Hall D

You can also stop by to see either Tom or Andy during the reading period week.  Either just stop by and see if someone is in - usually one of us will be in the office on any given afternoon, or send one of us an email to set up an appointment.

...and remember  to take advantage of the Math Question Center which meets from Sunday to Thursday from 8 to 10 pm in Loker.

Also be sure to read through the list of topics from the textbook that will not be included in this second midterm (these are located at the bottom of the list).  Also, anything that was specifically left off the list of topics from an earlier midterm (for instance section 3.4), will still be left off in terms of the final, so be sure to read through the earlier midterm review pages for that information as well.

Topics for final (in addition to those listed in the first two midterm review lists):

• Determinants:
• Definition of Determinant
• ability to calculate by hand for small matrices
• ability to use Gauss-Jordan elimination to calculate determinants (algorithm 6.2.6)
• ability to recognize when determinant is zero for larger matrices (linearly dependent column vectors, for instance)
• Basic properties of determinants
• determinant of matrix products
• effect on determinants of swapping rows, multiplying by scalars, etc.
• No information on Minors or Laplace Expansion (bottom of page 259 to 265)
• Nothing from section 6.3
• Eigenvalues, Eigenvectors and Eigenspaces
• Basic definitions of each concept
• an eigenvector for a linear transformation is one for which the transformation looks just like scalar multiplication
• the associated eigenvalue is the scalar involved - check definition 7.1.1 on page 294)
• Ability to find eigenvalues and eigenvectors for a given transformation
• to find eigenvalues find the roots of the characteristic polynomial (check fact 7.2.1 and 7.2.5)
• to find eigenvectors one needs to find the kernel of a the appropriate transformation (definition 7.3.1)
• Definition of algebraic and geometric multiplicity
• Relationships between eigenvalues for similar matrices (fact 7.3.8)
• Ability to work with complex eigenvalues
• although we didn't have homework from section 7.5 you should still be able to:
• find complex eigenvalues
• compute the modulus of a complex number
• add, subtract and multiply complex numbers
• know Euler's formula (fact 9.2.2)
• Eigenbases
• Definition - a basis for R(n) composed of eigenvectors for a given n by n matrix A (definition 7.3.4)
• Ability to check whether an eigenbasis exists for a given square matrix
• for instance an eigenbasis will definitely exist if the matrix's eigenvalues are distinct
• Relation between the existence of an eigenbasis for a square matrix and diagonalization
• in fact they are the same - fact 7.4.3
• know the definition of diagonalizable (7.4.2) and be able to diagonalize a matrix (i.e. find its eigenbasis and eigenvalues and create matrices S and D in definition 7.4.2)
• Discrete Dynamical Systems
• Use of eigenvalues to solve problems involving discrete dynamical systems
• Check the examples in section 7.1 and example 7 in section 7.3
• Be familiar with the approach outlined in the first part of fact 7.1.3
• Phase portraits - check the summary diagrams on page 299 and on page 420, which includes the continuous dynamical systems as well.
• Stability - section 7.6 - know what happens in the longterm for specific cases
• 0 is a stable equilibrium precisely when the modulus of all the eigenvalues is less than 1 (this includes complex eigenvalues - the "modulus" of a real number is just its absolute value)
• Although we didn't work through the details of example 3 in section 7.6, you should be able to simply use the result given in fact 7.6.4 to understand the longterm behavior for systems involving complex eigenvalues (again the pictures at the bottom of page 357 should be helpful).
• Symmetric Matrices - section 8.1 (the only section covered from chapter 8)
• Basically one punchline - given a symmetric matrix it's possible to create an eigenbasis for that matrix that is in fact orthonormal, and in the other direction if a matrix has an orthonormal eigenbasis then it must be symmetric.
• From the perspective of diagonalization (section 7.4) this is equivalent to saying that a matrix is orthogonally diagonlizable if and only if it is symmetric (fact 8.1.1 known as the Spectral Theorem)
• Continuous Dynamical Systems (chapter 9 - all three sections)
• Basic setup:
• Know how to deal with basic examples as illustrated in section 9.1
• Know the basic differential equation involved (page 397 - leading to the exponential function)
• Know the difference between a discrete and a continuous dynamical system (check bottom of page 399)
• Know how to solve a system given the existence of an eigenbasis (facts 9.1.2 and 9.2.3)
• Stability - similar to discrete case (fact 9.2.4), but now the question is whether the real parts of the eignevalues are negative
• Know the phase portrait comparisons between discrete and continuous dynamical systems (page 420)
• Linear Differential Equations (section 9.3)
• Basic definitions:
• Definition of a linear differential operator and linear differential equations (definition 9.3.1)
• Know fact 9.3.2 on the form of the general solution for a linear differential equation
• Characteristic polynomials
• know definition and be able to use it to find the kernel of a linear differential operator (fact 9.3.8)
• know the dimension of the kernel fact 9.3.3
• Be able to rewrite solutions involving complex roots of the characteristic polynomial (fact 9.3.9)
• Skip the operator approach to solving linear differential equations in section 9.3 (pages 432 to 435)
• Chapter 10 - there will be no questions from the handout on the final (i.e. no Fourier series or heat equation questions)
• Note there are several topics in the textbook that we did not cover in class, and which will not be covered on the final:
• No information on Minors or Laplace Expansion (bottom of page 259 to 265)
• Skip section 6.3
• Skip sections 8.2 and 8.3
• Skip the operator approach to solving linear differential equations in section 9.3 (pages 432 to 435)

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Textbook  Practice Problems:

Suggested practice problems:

Chapter 6:
Section 6.1 # 35, 37, 40, 45, 52
Section 6.2 #6, 8, 15, 23, 24
Solutions

Chapter 7:
Section 7.1 #36, 38, 39, 40
Section 7.2 #4, 8, 14, 15, 28
Section 7.3 #12, 20, 22, 25
Section 7.4 #16, 35, 36, 55
Section 7.6 #15, 28, 30, 40
Solutions

Chapter 8:
Section 8.1 #10, 15, 19, 21
Solutions

Chapter 9:
Section 9.1 #27, 28, 40, 42
Section 9.2 #9, 12, 16, 18, 26
Section 9.3 #8, 10, 12, 27, 28
Solutions

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Textbook  True/False chapter review problems:

• Another good way to get ready for the final is to do the true/false review problems at the end of each chapter that we've covered.   Note there are no True/False for chapter 9 or 10 (the handout), and we only covered one section from chapter 8, so the true/false questions that will be most useful for you are simply the ones from chapters 6 and 7.
• Note that although you should be able to do practically all of the questions from chapters 6 and 7, there are a few that come from topics that we've taken off the list, so there are a couple of problems that you shouldn't expect to be able to do - you should be able to figure out which ones these are by checking the list of topics covered/not covered, and by reading through the answers:

Solutions to Chapter 6 True/False problems
Solutions to Chapter 7 True/False problems