Review Sheet for the Final Exam, Fall 2015
Colorado State University, Pueblo
Math 207 — Matrix and Vector Algebra with Applications

Prior Material

New Terminology

1. eigenstuff for a matrix, being the
   eigen-
   • -values,
   • -vectors, and
   • -spaces
   of a matrix
2. the characteristic polynomial of a matrix
3. Markov models, particularly
   • the setup, as a system with
     • a finite number of states,
     • discrete time, coming with the ticking of a clock,
     • a transition matrix $T$ (which is stochastic) telling us with entry $t_{ij}$ the probability of the system transitioning from state $j$ to state $i$; which only really makes sense because of
     • the Markov hypothesis telling us that the transition probabilities are constant and depend only upon the current state of the system.
   • we talk about situations with $N$ parallel Markov systems (all with the same number of $n$ states and transition matrices), for which we have
     • a state vector (in $\RR^n$) telling us how many of the parallel systems are in each particular state,
     • the transition operation which tells what the state vector will be after one tick of the clock: $\vec{s}^\prime=T\vec{s}$, where $T$ is the transition matrix, $\vec{s}$ is the state vector before the tick, and $\vec{s}^\prime$ is the state vector after
     • a steady state vector, also called a stable distribution, being a distribution which remains unchanged when multiplied by the transition matrix

Part I of the Final

The material in this first part will be from what was covered on the first midterm (see its review sheet), together with questions on determinants and cross products. Specifically, there will certainly be problems like

• Doing arithmetic with vectors, using the geometric and component-wise approaches for vector addition, dot products, and projections. Find angles between vectors (e.g., knowing when two vectors are orthogonal), areas of triangles, lengths of projections, etc.
• Equations of lines and planes, in vector, parametric, and normal forms.
• Checking if something is a vector subspace of $\RR^n$
• Matrix arithmetic, such as multiplying different size matrices, knowing that multiplication is not commutative, using the identity matrix, etc.
• Checking if something is a linear transformation. If it is, finding the corresponding matrix.
• Working with rotations, reflections, and projections as linear transformations, e.g., finding their matrices.
• Computing determinants of $2\times 2$ and $3\times 3$ matrices. Using that to tell if the matrix is invertible or not.
• Computing the cross product of two vectors, and knowing to do that when you want a vector orthogonal to two others.

Part II of the Final

The material in this first part will be from what was covered on the second midterm (see its review sheet), except nothing will ask [explicitly] about determinants or cross products. In addition, there will be problems on the material covered since the second midterm, for which see the list of subjects above on this page. Specifically, there will certainly be problems like

• Solve a system with Gaussian or Gauss-Jordan elimination.
• Do row operations on a matrix, get it to REF and all the way to RREF.
• Find the inverse of a matrix.
• Use the inverse of a matrix to solve a system of linear equations
• Tell if a set of vectors is linearly dependent or independent.
• Tell if a given vector is in the span of some other vectors.
• Find the range and nullspace of a given matrix, and bases for those spaces.
• Compute the characteristic polynomial, eigenvectors, and eigenspaces (maybe with bases) of a given matrix.
• Explain why some situation does or does not qualify as a Markov model.
• Take a description of a Markov model, write down its transition matrix and state vector, figure out what the state vector will be some number of clock ticks in the future, and compute the steady state vector of the model.

Rules for Taking Both Parts

The total of your scores on both days will be your final exam grade, unless it is more than 200 in which case the excess will be kept as course extra credit and you will get 200 on the final.

You may — and indeed are expected to — bring a calculator to the final exam.

You may also, should you like, bring some notes to the final. Here are the details:

• Each day, you may bring one 8.5x11 page of paper, with equations written on one side only.
• This page must be handed in with the test at the end of that part of the test. [Therefore, it can be different material on the two days, and you should not rely on having Wednesday's page to prepare a page for Friday.]
• All that you may have on this page are equations. No words. No examples. This is not the place for explanations or your favorite example of some technique. It is a place for equations that you know and know how to use, but in which you worry that you might misplace a term.
• Any violation of the rules for this page of notes will cause a loss of 10 points per violation (i.e., per word or example).
• If you have any concerns about what may be allowed on these formula pages, talk to me before the test day so we can avoid any unpleasantness.

How to Prepare

Next, make sure you can do all assigned HW problems, all the mini- and maxiquizzes, and all of the problems from both midterms, without any trouble.

If all of that is really solid and you still have time ... maybe go for a walk? Read a good book? Learn to cook a soufflé? Do whatever you want, you are certainly prepared for your 207 final.

Getting Your Grade as Quickly as Possible after the Final