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 (turned in by the end of class) I suggest that you review your class notes and go over ASULearn solutions to the practice problems and problem sets. Here are the topics we have been focusing on:

> 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);

> 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:

There will be some short derivations - the same as we've seen before, like:

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,

1) If C is not invertible and AC=BC then A=B sometimes but not always. (True)

2) If C is invertible and AC=BC then A=B sometimes but not always. (False - it is always true since you can multiply by the inverse of C on the right.)

3) As long as the matrix mult is defined, then (A-B)(A+B) always equals A^2 - B^2. (False because A^2-BA+AB-B^2 is sometimes but not always the same as A^2-B^2, even for invertible matrices, by examples)

4) Matrix multiplication is always commutative (ie AB=BA) for square matrices. (False because it is sometimes but not always true by examples).

5) For any square matrices A and B of the same size, if AB = 0 then A=0 or B=0. (False there are lots of counterexamples).

6) If A is an invertible

7) An augmented matrix with a row of 0s has infinite solutions (False, this is true when the number of equations is less than or equal to the number of variables, because we would be missing a pivot in a spot, but it could be false via example when there are more equations than variables, like we have seen in clicker questions.

8) and more: others like we have seen in homework, clicker questions, problem sets...

You should also know some of the basic visualizations of rows of augmented matrices from Chapter 1 - that a linear system in two variables with one free variable [x+y=1] is a line, and that a linear system in three variables [x+y+z=1] with two free variables is a plane, as well as some of the visualizations of intersections of these to form solutions, like on ASULearn. Recall that once we solve an augmented matrix by reducing, the solutions tells us the geometry of the intersections of the rows: one free t variable means the rows intersected in a line, two free variables, like s and t, means the rows intersected in a plane, and no free variables arises from the rows intersecting in 1 solution, or from them being parallel, with no solutions. From the reduced matrix, you should also be able to write out the solutions quickly [inconsistent=0, 1 solution, or using free variables to write out infinite solutions in parametric form].