Dr. Sarah's Test 1 Study Guide

This test covers 1.1-1.3, 2.1-2.3 and portions of 2.5, and 3.1-3.3 and portions of 3.4. 1 8.5 x 11 sheet with writing on both sides allowed. You may put anything you want that fits on your sheet, including examples, statements, solutions to problems, or Maple commands. You will be asked to show by-hand steps on some parts, Maple on others, and will be allowed to use your calculator (if you bring one) or Maple on other parts. Unlike the problem sets, you will not need to print out calculator or Maple work - just hand write the reduced form or relevant matrices on your test.

Be sure to study Solutions to Problem Sets 1-3 that are up on ASULearn, along with the practice problem solutions, and make sure that you understand the related material (statements of theorems, examples of false statements, ...) very well.

  • You are responsible for knowing the big picture methodology/approach of the following proof from class, that you could describe in a couple of well chosen sentences, but don't worry about all of the proof details.
    --The number of solutions of a system of n linear equations in n unknowns [ie an n x (n+1) augmented matrix] has no solutions, exactly 1 solution, or an infinite number of solutions.
    Specifically, what were the examples we used [with "dots" in them to account for the number of rows being unknown n] that showed 0, 1, or infinite solutions.
    In addition, how did the heart of the proof work? Ie how did we show that if we started with 2 solutions, then really we had infinitely many solutions that we generated?

  • You should be very comfortable with writing out an augmented matrix for a system and using Gaussian Elimination by-hand, or ReducedRowEchelonForm(A); on Maple in order to solve it. 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].

  • You should also be comfortable with by-hand matrix multiplication and determinants of 2x2, 3x3 and 4x4 matrices.

  • You can expect to be given some problems similar or the same as those on the problem set solutions. For example, you can expect a k, or a, b, c unknown in the matrix kind of problem, like on problem set 1, and a problem like the Healthy/Sick workers on problem set 3, including the method of solving by taking (M-Identity matrix)x = 0, getting a solution, and setting the sum of the solution=1 to preserve population [see the demos and solutions on ASULearn - for example, one shortcut was to put a row of 1s at the bottom of the augmented matrix], to determine stability, as well as using only the definition of a stochastic regular matrix to determine whether a system will stabilize (do the columns add to one and are the entries all positive?).

  • You can expect to be given statements and to either have to state that they are true (no need to say why - just say "true"), to correct a false statement, or to provide a counterexample to show that they are false, depending on what the directions say.

    For example, you might be given any of the true/false statements, such as:
    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 nxn matrix, and x and b are nx1 vectors, then the matrix-vector equation A*x=b has a unique solution. (True)

    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, or write these down on your cheat sheet.

    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.

    Some Test Directions

    Be careful to follow all directions and to read each problem very carefully to make sure you answer correctly. You do NOT need to write in complete sentences like you do on the problem sets. Show work and briefly explain where indicated. Even if it is not indicated, you may wish to do so because partial credit can be obtained for an incorrect answer as long as there is some evidence of understanding. You may use your calculator or Maple unless the problem says "by hand". When you are finished, turn in your cheat sheet with your test. You will not turn in any Maple work.

    Some Maple Commands Here are some Maple commands you should be pretty familiar with by now for this test - you may wish to put these on your cheat sheet.
    > with(LinearAlgebra): with(plots):
    > A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
    > ReducedRowEchelonForm(A);
    > GaussianElimination(A);
    (only for matrices with k or a, b, c in the augmented array.)
    > B:=MatrixInverse(A);
    > A.B;
    > A+B;
    > B-A;
    > 3*A;
    > A^3;
    > evalf(A^100);
    or evalf((A^100).Initial); (be careful to use fractions for stochastic matrices)
    > 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);