Test 1 Study Guide: Selections from 1.1-1.5, 1.7, 2.1-2.3 and applications

It is time for our first test in order to be sure that everyone reviews and understand some of the fundamental concepts.

At the Test

  • You may make yourself some reference notes on the small card I hand out (additional cards are on my door if you need to rewrite it). The reference card must be handwritten. Think of the card as a way to include some important examples or concepts that you aren't as comfortable with. You won't have room for everything, and you should try to internalize as much as you can.
  • A calculator will be allowed (but no cell phone nor other calculators bundled in combination with additional technologies) but is not required.
  • You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be standalone--no cell phone, internet or other technological connections)
  • 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.
  • 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 is 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 typeset professionally (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." Some students in the past have reported that they found it helpful to go through the sample partial test, review other problems we covered, skim solutions, and read through the glossary on ASULearn. Videos are also available to help you review. I'm happy to help you in office hours, Zoom, or on the ASULearn forums--for example, you can bring in the sample test and go through it with me, or ask me any other questions you have. We'll also spend part of the class before the test on some review activities, and I will take some class time to answer any questions.

    Problems Covered

    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 critical thinking and problem-solving skills (I'm trying to help you develop your linear independence!), so I suggest that you review those solutions on ASULearn and from notes you took during class. I am happy to help you with any related material you need to brush up on.

    worksheet on Evelyn Boyd Granville's Favorite Challenge
    1.1 1, 3, 14, 19, 21, 23, 24 part d, 25, 31
    1.2 3, 7, 19, 11, 21e, 25, 30, 31, 32
    clicker questions in 1.1 and 1.2
    problem set 1
    glossary for 1.1 and 1.2
    1.3 1, 5, 9, 15, 23 b,c,e, 27 a, b, 30
    1.4 3, 9, 13, 15, 23 b,f, 31, 33
    1.5 3, 5, 9, 13, 19, 21, 23 a, d
    1.7 1, 3, 9, 21 b, d, 22 b, 29, 36
    worksheet on 1.3-1.7
    clicker questions in 1.3-1.7
    problem set 2
    glossary for 1.3-1.7
    2.1 5, 9, 10, 15 b, e, 18, 21, 23, 27
    2.2 1, 5, 9c, 12, 13, 17, 21, 23
    2.3 3, 5, 11 c & d, 12 c & e, 19, 21, 23, 43
    clicker questions in 2.1-2.3
    worksheet on definitions
    problem set 3
    glossary for 2.1-2.3

    Topics Covered

    review slides
    class highlights page
  • 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 intersecting in a point, line or plane according to the number of free variables in a parametrization
  • homogeneous systems and their solutions
  • equations meeting certain criteria 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, i.e. 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, i.e. full column pivots]
  • span but not linearly independent; linearly independent but not span
  • practical applications of span; of linearly independent
  • Algebra of matrix multiplication [the definitions of AB, the number of multiplications in AB]
  • Elementary Matrices like Matrix([[1, 0, 0], [0, 1, 0], [-2, 0, 1]]) representing the row operation r_3' = -2r_1 + r_3 or Matrix([[1,0],(-2,0]]) representing the row operation r_3' = -2r_1 + r_2
  • Inverse of a Matrix as a concept A.A-1=I and in Maple and the computational formula for the inverse of a 2x2 matrix
  • Transpose of a Matrix
  • Matrix algebra properties that do hold, often reasoned using some combination of
    applying an inverse (if it exists) to both sides of an equation,
    reorder parenthesis by associativity, cancel A by its inverse, and
    reducing the identity to get whatever is left alone.
    [like Ax=b has 1 unique solution x=A-1b when A-1 exists and x and b are the correct size column vectors...]
    OR reasoned using arguments involving pivots/missing pivots
  • Matrix algebra properties that don't hold and counterexamples [like 2 non-zero matrices that multiply to yield a zero matrix, AB is not necessarily BA...]
  • Square matrix theorem (last slide of What Makes You Invertible): The following are equivalent for square matrices A_nxn:
    A invertible, A reduces to the Identity
    matrix, columns A span Rn, A has full row pivots,
    columns A linearly independent, A has full column pivots,
    Ax=b has 1 unique solution x=A-1b,
  • Negations of the square matrix theorem for non-invertible A_nxn matrices, like a square nxn matrix A that does not row reduce to the identity matrix is logically equivalent to the columns of A not spanning Rn.
  • Examples and counterexamples of the square matrix theorem when A is not a square matrix [like examples of a matrix whose column vectors span but are not l.i...]
  • Hill Cipher: coding using A.uncoded message and decoding using A-1.coded message
  • Condition Number [you do NOT need to know the formulas - just the big picture idea]
  • Decimals versus fractions in a computer algebra software program like Maple
  • 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)
    > B:=MatrixInverse(A); ConditionNumber(A);
    (only for square matrices)
    > Vector([1,2,3]); Transpose(A); A+B; B-A; 3*A; A^3; 3*S + 5*A + 2*L; A.B;
    > 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 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 (i.e. 3 planes intersect in a point, a line, a plane, or no common points)