### Test 3 Study Guide

This test will be closed to notes/books, but a calculator will be allowed (but no cell phone nor other calculators bundled in combination with additional technologies). There will be various components to the test and your grade will be based on the quality of your responses in a timed environment (must be turned in by the end of class). Fill in the blank
Calculations and Interpretations
Short Derivations
True/False

Be sure to study tests 1 and 2 and any related material that you need to brush up on (test 3 will be comprehensive), and PS 6 solutions along with the recent activities and demos on the class highlights page and ASULearn. In the book, portions come from 7.1, 7.2, portions of 7.4, and some of chapter 6.

Specifically, here are the topics we have been focusing on:
• test 1 and 2 material (be sure to study test 1 and any related material that you feel that you need to brush up on)
• algebra and geometry of eigenvectors [Ax=lambdax, (lambdaI-A)x=0, matrix multiplication turns to scalar multiplication for eigenvectors, so they are vectors that are on the same line through the origin]
• number of eigenvectors, number of linearly independent eigenvectors
• basis of eigenvectors
• diagonalizability
• linear transformations of the plane and their corresponding eigenvalues and eigenvectors (projections, dilations, reflections, rotations,and shear matrices)
• revisiting previous problems - healthy and sick workers, Brand A, B, neither, and understanding them at a deeper level
• eigenvector decomposition and what happens in the limit (directions, ratios...) for various initial conditions (If a2=0... otherwise we die off, grow, stabilize...) including the equations of the lines we come in along.
• geometry of stability situations [for example, if xk = a1 1k Vector([2,1]) + a2 .7k Vector([-1,1]) then as long as a1 is non zero, we will stabilize to the y=1/2 x line via the populations ratio of 2:1. Graphically you should be able to draw pictures like in the Dynamical Systems demo - see the second last picture in the demo for this situation. You can tell from the algebra that given a starting position, you will come in parallel to Vector([-1,1]) (i.e. x+y=1) until we eventually hit the stability line, where we stay forever, and that the contribution from Vector([-1,1]) is smaller and smaller with each k, which is also represented in the picture.
• fractions in Maple, versus errors with decimals in Maple
• Maple output of eigenvectors giving one basis representative for each line, or 2 basis representatives for each plane.

In addition to the above, from the ASULearn solutions and your old tests, in addition to reviewing them in general, also be sure to carefully go over
• PS 1 k or a,b,c problem.
• PS 3 Healthy/Sick Workers and Practice Problem Brand A, Brand B, Neither matrix: setting up stochastic matrices and stability as eigenvalue of 1
• PS 4 and PS 5 cement mixing problems,
• PS 6 rotation matrix eigenvalue problem,
• PS 6 fox problem
• Summary of Equivalent Conditions for Square Matrices on p. 239 (don't worry about Rank as we did not cover this.)
• vector space axioms 1 and 6 and their negations (ie the subspace axioms),
• definitions and examples of basis, linear independence and span
• definition and examples of eigenvalue and eigenvectors

You can expect to see problems which are similar to these and/or problems that begin with a question like you've seen in an earlier section, but use some of our recent language to explore further.

You can also expect to have a problems which ask you to give examples, such as
• examples from test 1 and 2, such as a matrix A and vector b so that Ax=b has a certain number of solutions, and span and li examples in R2 and R3 (see test 2 study guide for comments on span and li along with test 2).
• examples of 2x2 matrices which have a certain number of real eigenvalues (0,1,2) and eigenvectors (0 or infinitely many) or linearly independent eigenvectors (0,1,2).

For example, if I asked you to produce an example of a matrix with 1 eigenvalue, and 1 linearly independent eigenvector, then Matrix([[1,0],[1,1]]) would work (notice though that it has infinitely many eigenvectors that are not linearly independent, because constant multiples of an eigenvector produces other eigenvectors, as anything on that same line through the origin still stays on that line. This is a vertical shear matrix with just the y-axis having eigenvalue 1. A basis for the line can be represented by [0,1]).

For 2x2 matrices, we cannot have more than 2 linearly independent eigenvectors because they would form a basis for R^2, which has at most 2 linearly independent vectors in a basis, [but we can find examples of 3x3 matrices that have exactly 3 linearly independent eigenvectors]. Using geometric intituion can help quite a bit for problems too. For example, we can use the geometry of a rotation, projection, etc, in order to explain what (if any) the eigenvalues and eigenvectors are, and to generate examples quickly, as we discussed in class, and you should know these from class notes. Review why most rotation matrices have no real eigenvalues and eigenvectors (same line through the origin arguments), why a projection matrix has an eigenvalue 1 corresponding to [cos(theta), sin(theta)] and an eigenvalue 0 corresponding to [-sin(theta), cos(theta)], etc.

Also think about some broader connections of the class - applications that we have covered, algebraic and geometric perspectives, how calculus has played a role, some of the historical perspectives...

Some Maple Commands Here are some Maple commands you should be pretty familiar with by now for this test - i.e. I will at times show a command, and it may be with or without its output:
> with(LinearAlgebra): with(plots):
> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
> ReducedRowEchelonForm(A);
> GaussianElimination(A);
(only for augmented matrices with unknown variables like k or a, b, c in the augmented matrix)
> B:=MatrixInverse(A);
> A.B;
> A+B;
> B-A;
> 3*A;
> A^3;
> evalf(A^100);
or evalf((A^100).U); (be careful to use fractions for stochastic matrices)
> Determinant(A);
> Vector([1,2,3]);
> Eigenvectors(M);
> Eigenvalues(M);
> evalf(Eigenvectors(M));
> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2);
plot vectors as line segments in R3 (columns of matrices) to show whether the the columns are in the same plane, etc.
> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);
> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4);
plot equations of planes in R^3 (rows of augmented matrices) to look at the geometry of the intersection of the rows (ie 3 planes intersect in a point, a line, a plane, or no common points)

There will be some fill in the blank short answer questions, such as providing:
• definitions related to any of the above topics, including test 1 and 2 material
• real-life applications, like ____ is a real-life application of matrix inversion (where the natural answer would be the Hill cipher)
• fill in the blank related to computations, examples and interpretations

• There will be some by-hand computations and interpretations, like those you have had previously for homework, clicker questions and in the problem sets.

### Derivations

There will be some short derivations - the same as we've seen before, like:
• Those on test 1 study guide and test 2 study guide
• Derivation that solving Ax=lambdax is equivalent to solving the system (lambdaI-A)x=0
• Derivation that for eigenvectors x for A, Akx = lambda kx
• Derivation that A P = P times the diagonal matrix of eigenvalues [which is how we showed that MatrixInverse(P).A.P = Diag]

### True/False

True/False statements and counterexamples has been a recurring theme in all of the chapters, so you can expect problems like we have seen before in practice problems and problem sets. Be sure that you know how to find counterexamples There will be questions were you answer true or false and if false, then you will either correct the text after the word "then" (that does not change equal to not equal for ex) or provide a counterexample.
For example,
• It is possible to find a matrix with 2 linearly independent eigenvectors
• It is not possible to find a 2x2 matrix with 3 linearly independent eigenvectors
• It is false that eigenvectors allow us to turn matrix multiplication into addition [should be scalar multiplication]
• algebra and geometry of objects like Gaussian reductions like t (row 1) + (row 2) [parallel to row 1 through the tip of row 2 and the operation preserves area or volume which is the determinant] or linear combinations like c(column 1) +d (column 2) [a plane through the origin if the columns are l.i. and a line through the origin otherwise].
• Others mentioned above before the Maple commands