Specifically, here are the topics we have been focusing on:

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

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

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...

> with(LinearAlgebra): with(plots):

> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);

> ReducedRowEchelonForm(A);

> GaussianElimination(A);

> B:=MatrixInverse(A);

> A.B;

> A+B;

> B-A;

> 3*A;

> A^3;

> evalf(A^100);

> 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);

> 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);

There will be some fill in the blank short answer questions, such as providing:

For example,