Test 2 Study Guide (cumulative + 2.1-2.3, 1.8, 1.9, 2.7 and applications)

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:
  • Definitions
  • Calculations and Interpretations / Analyses
  • Conceptual 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." I suggest that you review test 1 and any related material that you feel that you need to brush up on, as this test is cumulative. In addition, review your class notes from the material since test 1 and go over ASULearn solutions to the practice exercises and problem sets as well as the clicker questions. Questions will be typeset using LaTeX (like this) and will typically be formatted as follows:
  • Fill in the Blank
  • Short Answer Calculations and Interpretations / Analyses
  • True/False Questions: You may be asked to identify true statements [no need to recall page #s nor Theorem #s like in problem sets], provide counterexamples, and/or correct statements
    • Here are the topics we have been focusing on:
  • Test 1 material [test 1 study guide] including definitions such as linearly independent, span... Test 2 will be a majority of new material but still have a portion of test 1 material.
  • 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
  • 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 appling an inverse (if it exists), performing associativity, and reducing using the definition of the inverse and the identity, [like B-C=0 implies B=C, or Ax=b has 1 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:
    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 solution x=A-1b, the linear transformation T(x)=Ax is has range all of Rn...
  • 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...]

    Applications:
  • 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]
  • Linear Transformations of the plane, both 2x2 as well as 3x3 homogeneous coordinates versions. Know the following:
    general rotation matrix
    projections onto the y=x line, and the x and y axes
    reflections across the y=x line, and the x and y axes
    horizontal shear
    dilation
    translation
  • Composition of linear transformations: right to left (ABCx means first C(x) then B applied to that then A applied to that result, just like with function composition)
  • Rotate about a point, like (5,9): Translate by (5,9).Rotate.Translate by (-5,-9)
  • Big picture ideas of Yoda and transpose of a matrix
  • Computer speed of (AB).Large matrix versus A(B.Large matrix) including counting the number of multiplications and reasoning that (AB).Large matrix is much faster
  • 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 a, b, c in the augmented matrix)
    > Transpose(A);
    > ConditionNumber(A);     (only for square matrices)
    > Vector([1,2,3]);
    > B:=MatrixInverse(A);
    > A.B;
    > A+B;
    > B-A;
    > 3*A;
    > A^3;
    > 3*S + 5*A + 2*L;
    > evalf(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)     Some Sample Test Questions
  • See the sample test questions for test 1 at the bottom of the test 1 study guide, as well as your test 1 itself. Like test 1, almost all of test 2 will be from the practice hw, clicker questions and problem sets, as well as definitions, big picture ideas or computations that are listed on either study guide.

    Fill in the blank
  • The definition of A invertible is_________ (there is a B so that AB=Identity)
  • Matrix([[-1,2,1],[0,0,1]]).Matrix([[1,2,3],[0,1,0],[1,0,1]])= _________
  • Transpose means_________
  • The definition of linearly independent is __________
  • The definition of span is _________
  • To satisfy the definition of span, we can have _______ solution(s) (1 or infinite)
  • To satisfy the definition of l.i., we can have _______ solution(s) (1, just the trivial solution (0 vector))
  • An example of vectors that span R2 but are not l.i. is _______ (Vector([1,0], (Vector([2,0]), (Vector([1,1]))
  • An example of vectors that are l.i. in R2 but don't span R2 is _______ (Vector([1,0])
  • An example of 2 vectors that span R2 but are not l.i. is __________ (not possible by the square matrix theorem)
  • ________ is a real-life application of a linear combinations of vectors (center of gravity, cement mixing)
  • ________ is a real-life application of the inverse of a matrix (Hill cipher)
  • ________ was working with 6x6 matrices (Hill, with his "mechanical message protector")
  • ________ is a real-life application of the transpose of a matrix (Digital Yoda where data was in the the rows instead of the columns, linear regression in statistics)
  • If the condition number is on the order of 104 then that tells us that we may lose up to _________ (4 digits of accuracy: it measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument.)
  • Maple 1.0 dates back to ________ (the computer revolution of the 80s, or 1982)

    Computations and Interpretations/Analysis:
  • Just like on test 1, in this section the questions will come from questions you have seen before in practice problems, the problem set, clickers (and now) test 1 (for example, might now be rephrased using some of our recent language).

    True/False
    Like on test 1, these are similar to practice problems, problem set or clicker questions (and now) from test 1 too. They will (again) either be formatted as "circle true or correct after the word..." or "circle true or provide a counterexample" (or could show up as fill in the blanks too). Here I have given you samples that connect to how many solutions a system can have, just for review sake:
  • If A is an nxn matrix, and x and b are nx1 vectors, then the matrix-vector equation Ax=b has 0, 1 or infinite solutions. (True)
  • If A is an nxn matrix, and x and b are 1xn vectors, then the matrix-vector equation Ax=b has 0, 1 or infinite solutions. (False - just 0 solutions as the multiplication does not make sense)
  • If A is an invertible nxn matrix, and x and b are nx1 vectors, then the matrix-vector equation Ax=b has 1 solution (true x=A-1b).
  • If A is an nxn matrix that is not invertible, and x and b are nx1 vectors, then the matrix-vector equation Ax=b has 1 solution (false 0 or infinite. Since the coefficient matrix is not invertible then A can't reduce to the identity matrix. Since the matrix IS square, then the last row in Gauss-Jordan must be all 0s and so we'll be missing a pivot for at least one variable. If the equal column gives us a consistent system we'll have infinite solutions, and if not 0 solutions).
  • If A is not square then the matrix-vector equation Ax=b can't have just 1 solution for some b (false - think of 3 lines interesecting at just one point, for example).
  • If A is not square then the matrix-vector equation Ax=b can't have 0 solutions for some b (false - think of 2 parallel planes, for example)
  • If we have a system of less equations than unknowns then we'll have 0 or infinite solutions (true - we won't be able to have a pivot for each variable. We might have 2 non-parallel planes, that would always intersect in a line, or 2 parallel planes that would either overlap or be inconsistent depending on the equal column)
  • Ax=0, with A mxn, x nx1 and 0 mx1 can have 0, 1 or infinite solutions. (false - correction: 1 or infinite solutions (as the trivial (0 vector solution for the weights) always works)