Study Guide for Test 2

1 8.5 x 11 sheet with writing on both sides allowed. You may put anything you want that fits on your sheet. Calculator allowed, but not necessary. You will be asked to show by-hand steps, Maple on others, and will be allowed to use Maple or a calculator on other parts. I view this exam as a learning experience. This means that it is not only an opportunity for you to demonstrate your mastery of the material in a timed setting (the test must be turned in by the end of class), but it is also an opportunity for you to make connections.

Be sure to study test 1 and any related material that you feel that you need to brush up on, and ps 4 and ps 5 solutions on ASULearn, including the cement problems, and group work from class (solutions are on ASULearn). This test will mostly concentrate on ideas in Chapter 4 (4.1, parts of 4.2 related to matrices and Rn, 4.3, 4.4, 4.5, and selections of 4.6 relating to solutions of homogeneous systems or solutions of systems of linear equations (we didn't cover elementary matrices back in chapter 2.4, so we didn't cover the rank or row space here), but will include some material from test 1. Be sure that you understand the material and can use this understanding to quickly answer questions.

  • You need to know why things are or are not vector spaces and subspaces, and know the reasons and relevant counterexamples, but you will not be expected to write out complete proofs. 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). You should be familiar with writing out what violating axioms 1 and 6 means, like there exists u, ... and following this to come up with specific counterexamples.

    You will also be asked to find counterexamples, but if no examples exist, then to say so. 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 [the shortcut theorem says if you have the correct number of elements for a basis 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 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.
  • 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 - (1,1) or any non-zero vector will do, but not 2, 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,
  • 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].

    You should review the cement problems carefully. In addition, be able to explain why a matrix with a non-zero determinant has columns that form a basis for Rn (where n is the number of rows/columns), know the definition and practical applications of span and linearly independent, and be able to apply these to problems, write out a basis for the corresponding homogeneous solution space for a system, and know the difference between the geometry of the columns of a matrix, and the geometry of the rows of an augmented matrix [many solutions on ASULearn present the various methods and geometry visualizations - so ASULearn is a good review of this - for example, see Problem Set 4 Solutions].

    Some examples of the kind of instructions that are likely to be on the test:
  • Write down examples that satisfy the following - there is no need to justify your examples, but if no examples exist, then do briefly explain why it cannot exist.
  • Which of the following are Vector Spaces? If it is a vector space, then just say so. If not choose a vector space axiom that is violated, write down Part 1: what it means to violate the axiom, and Part 2: why it is violated in this case via the appropriate examples, but no need for a complete proof, introduction, conclusion, nor complete sentences.

    Some Maple Commands Here are some Maple commands you should be pretty familiar with by now for this test - you may wish to put these on your cheat sheet.
    > 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)