Test 1 Study Guide: Selections from 1.1-1.5, 1.7 and applications

It is time for our first test in order to be sure that everyone reviews and understand some of the fundamental concepts. 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 types of questions on the test and your grade will be based on the quality of your responses in a timed environment (it will end 50 minutes after class begins and then we will resume class - if you are done early, you may leave and come back then).

For the first test only I have provided a sample partial test and solutions, so that you can see an example of the formatting and style of questions. As listed there you will see three sections that are typset formally (using LaTeX):

  • Fill in the blank
  • Computations and Interpretations / Analyses
  • True/False Questions
    As such the test will be a mixture of computational and definition questions as well as critical reasoning and questions involving the "big picture." Most questions are adapted from or taken right from exercises we had for homework, problem set questions, and clicker questions, although they may be rephrased or repackaged to further develop develop critical thinking and problem-solving skills, so I suggest that you review those solutions and any related material you need to brush up on. Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade. Here are the topics we have been focusing on:
  • algebra and geometry of equations and their solutions; algebra and geometry of rows of a matrix
  • Gaussian and Gauss-Jordan methods and history
  • augmented matrix, coefficient matrix
  • pivots/leading 1s
  • unique, 0 or infinite solutions algebraically and geometrically
  • parametrization of infinite solutions
  • lines or planes interesecting in a point, line or plane according to the number of free variables in a parametrization
  • homogeneous systems and their solutions
  • equations meeting certain critera and their solutions or consistency [like 2 equations in 3 unknowns, 3 equations in 2 unknowns...]
  • algebra and geometry of vectors; algebra and geometry of columns of a matrix
  • algebra and geometry of objects like Gaussian reductions like
    t (column 1) + (column 2) [parallel to column 1 through the tip of column 2]
    c(column 1) +d (column 2) [a plane through the origin if the columns are l.i. and a line through the origin otherwise, unless the columns were the trivial 0 vector].
  • writing out the solutions of a system as a vector parametrization equation with homogeneous plus particular portions
  • diagonal of a parallelogram, scaling along a line
  • scalar multiplication, addition and matrix multiplication of matrices and vectors (and relationship to systems of equations)
  • algebra and geometry of linear combinations and weights; mixing problems
  • applications of linear combinations to manufacturing, physics...
  • do we span R2, R3,... [setting up = general vector and obtaining never inconsistent, ie full row pivots]
  • what do we span? [line, plane, hyperplane...]
  • is a vector in the span?
  • linearly independent [setting up = 0 vector and obtaining only trivial solution, ie full column pivots]
  • span but not linearly independent; linearly independent but not span
  • practical applications of span; of linearly independent
  • 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 b1, b2, b3 in the augmented matrix)
    > Vector([1,2,3]);
    > 3*S + 5*A + 2*L;
    > evalf(M)
    decimal approximation of 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);
    > display (a,b,c);
    > 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)