Test 2 Study Guide

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
I suggest that you review your class notes and go over ASULearn solutions [since ASU is down I have listed some of the recent solutions here:] to the practice problems and problem sets.
  • Prob Set 4
  • Prob Set 5
  • Practice 4.4-4.6

    Here are the topics we have been focusing on:
  • test 1 material (be sure to study test 1 and any related material that you feel that you need to brush up on, including the derivations there)
  • algebra and geometry of vectors
  • algebra and geometry of linear combinations
  • algebra and geometry of row operations and determinants
  • vector spaces
  • linearly independence
  • span
  • basis
  • dimension
  • algebra and geometry of columns
  • algebra, geometry and basis of homogeneous solution spaces
  • writing out the solutions of a system as a homogeneous plus particular portion.
  • mixing problems

    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)
    > B:=MatrixInverse(A);
    > A.B;
    > A+B;
    > B-A;
    > 3*A;
    > A^3;
    > evalf(A^100);
    or evalf((A^100).U); (be careful to use fractions for stochastic matrices)
    > Determinant(A);
    > Vector([1,2,3]);
    > 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);
    > 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)

    Fill in the blank
    There will be some fill in the blank short answer questions, such as providing:
  • definitions related to any of the above topics, including test 1 material
  • 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

    Calculations 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. For instance, review the cement problems from ps 4 and 5 carefully, in addition to the practice problems, like the Maple practice problem.

    Short Derivations
    There will be some short derivations - the same as or similar to what we have seen before, like:
  • Those on test 1
  • Why things are or are not vector spaces and subspaces, and know the reasons and relevant counterexamples of axioms 1 and 6 (the addition and scalar multiplication axioms). If n is given as the number of rows or columns, but you are asked to leave n general, be sure to do so (ie do not define n to be 2 or 3 - instead, use the general nxn identity matrix, or the nxn matrix with all 0s except a 1 in the top left corner, or something similar for your examples), like in ps 4 solutions.
  • The proof that in a basis, each vector can be written as a unique linear combination of the basis vectors.
  • The proof that the only subspaces of R2 are the 0 vector, lines through the origin, and all of R2

    True/False 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, problem sets (solutions on ASULearn), and the i-clickers (questions on the class highlights page). 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 a given word like "then" or provide a counterexample.
    For example,
  • It is possible to find a set of 4 vectors that span R3 but are not linearly independent [make your life easy and choose a simple counterexample, like the standard R3 basis (1,0,0), (0,1,0), (0,0,1) which already spans R3 but add in the (0,0,0) vector or any other vector to get rid of efficiency)],
  • It is not possible to find a set of 3 vectors that span R3 but are not linearly independent [a theorem says if you have the correct number of elements for a basis in a known vector space with a given dimension, then span is true iff linearly independence is]
  • It is not possible to find a set of 2 vectors that span R3 [one needs 3 vectors to represent R3 - 2 non-trivial vectors can at most span a plane (if they are linearly independent) or a line (if they are linearly dependent multiples of each other)].
  • It is possible to find a set of 2 vectors that are linearly independent in R3 but do not span R3, by taking 2 vectors that span a plane in R3.
  • If a vector w is a linear combination of vectors in V, then adding w to those vectors will force the set to be not li (w=c1v1+... means that 0 = -w+ c1v1+... so there is a nontrivial solution to the homogeneous equation). On the other hand we have seen examples where w is not a linear combination of vectors in a set, but the set is not li, because one of the other vectors is a linear combination.
  • You should also be able to answer questions like this for R2 also! For example - you can find 1 linearly independent vector in R2 that does not span R2 - (1,2) or any non-zero vector will do, which spans a line with the given slope 2=(y-0)/(x-0). However 2 li vectors in R2 would not be possible, since two vectors in R2 that are linearly independent will also span.
  • an example of a set that spans R2 but is not linearly independent,
  • an example of 2 different basis sets for R2,
  • an example of a set that is li in R2 but does not span,
  • subspaces of R2 [0 vector, lines through origin, R2],and R3 [0 vector, lines through the origin, planes through the origin, R3],...
  • examples that show matrix multiplication is or is not commutative (like rotation matrices that are commutative),
  • examples of systems that have 0, 1 or infinitely many solutions to them,
  • algebra and geometry of objects like Gaussian reductions like t (row 1) + (row 2) [parallel to row 1 through the tip of row 2 and the operation preserves area or volume which is the determinant] or linear combinations like c(column 1) +d (column 2) [a plane through the origin if the columns are l.i. and a line through the origin otherwise].
  • and more: others like we have seen in homework, clicker questions, problem sets...