Test 1: 1.1-1.3, 2.1-2.3 and portions of 2.5, and 3.1-3.3

It is time for our first test in order to be sure that everyone reviews some of the fundamental concepts before we move on to Chapter 4.

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) Fill in the blank
Calculations and Interpretations
Short Derivations
True/False 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:
  • systems of equations and their solutions via algebra and geometry
  • Gaussian and Gauss-Jordan method of solution
  • inverse method of solution
  • parametrization of solutions
  • pivots/leading 1s
  • unique, 0 or infinite solutions
  • solutions of points, lines, planes
  • scalar multiplication, addition and matrix multiplication of matrices
  • other algebra of matrices
  • inverse matrix
  • determinants
  • Laplacian expansion of determinant by minors and cofactors
  • traffic problems
  • Hill cipher
  • population dynamics
  • digital image processing

    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 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);

    There will be some fill in the blank short answer questions, such as providing:
  • definitions related to any of the above topics
  • 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. You should be comfortable with matrix multiplication, Gaussian Elimination, and determinants of 2x2, 3x3 and 4x4 matrices by-hand. You can expect a k, or a, b, c unknown in the matrix kind of problem, like on problem set 1, and/or a problem like the Healthy/Sick workers on problem set 3, including the method of solving by taking (Identity matrix - M)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?).

    There will be some short derivations - the same as we've seen before, like:
  • What were the examples we used [with "dots" in them to account for the number of rows being unknown n] that showed 0, 1, and infinite solutions in the proof that 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?
  • How did we show that if we started with 2 solutions, then really we had infinitely many solutions that we generated in the proof that 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?
  • In population dynamics, what are the steps and reasons that show that (I-N)x=0 is equivalent to solving for stability.
  • At the end of 3.3, what are the ways we justify that one of the equivalent conditions implies the other (for example, if an inverse exists, why is there just one solution to a linear system).

    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 nxn matrix, and x and b are nx1 vectors, then the matrix-vector equation A*x=b has a unique solution. (True)
    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].