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Need some tough Linear Algebra Problems

  1. Jan 25, 2010 #1
    Hi all, I don't know if this is the correct place to ask this, but I am looking for some tough linear algebra problems (though still accessible to bright 1st years) to give to my class for possible extra credit.

    Any problems, or sources would be appreciated.
    Thanks!
     
  2. jcsd
  3. Jan 25, 2010 #2
    Here are two problems that were given to my freshman linear algebra class as optional hard problems:

    Problem 1 (given after a discussion of determinants in week 3/4 of the course):
    Consider a 9x9 matrix A. We say that A is a Sudoku matrix if it's the valid solution to a Sudoku puzzle. That is if,
    1) Every row and every column is a permutation of {1,2,3,4,5,6,7,8,9}.
    2) If we write it in block form:
    [tex]A = \left[\begin{array}{c|c|c} A_1 & A_2 & A_3 \\ \hline A_4 & A_5 & A_6 \\ \hline A_7 & A_8 & A_9 \end{array} \right][/tex]
    where [itex]A_i[/itex] is a 3x3 matrix, then every [itex]A_i[/itex] has elements {1,2,3,4,5,6,7,8,9}.
    Now the problem is:
    a) Find a Sudoku matrix with determinant 0.
    b) Does there exist a Sudoku matrix with determinant 1. If not then determine the least positive number that a Sudoku matrix can have as a determinant.

    Problem 2 (Given after discussing vector spaces, subspaces, linear independence, etc. in week 5/6. This was done as a sort of a contest where the professor picked out the best solutions which is possible due to discussion required in some of the problems.):
    The zeros of a real polynomial P(x,y) in two variables is called an algebraic curve. Let p(t), q(t) be real polynomials, and define a real function [itex]f : \mathbb{R} \to \mathbb{R}^2[/itex] by f(t) = (p(t),q(t)).
    a) Prove that the image of f, i.e. the set [itex]\{f(t) | t \in \mathbb{R}\}[/itex], is contained in an algebraic curve (different from the trivial [itex]\mathbb{R}^2[/itex] given by P(x,y)=0; in all subsequent questions we shall assume that P(x,y)=0 isn't a valid solution).
    b) Let [itex]p(t) = t^2[/itex], [itex]q(t) = t^3[/itex] and find a real polynomial P(x,y) in two variables such that the image of f is contained in the algebraic curve determined by P. Do the same question for [itex](p(t),q(t)) = (t^2+t,t^3)[/itex] and [itex](p(t),q(t)) = (t^2+t,t^2)[/itex].
    c) Consider how to define the degree of a polynomial in 2 variables [at this point we hadn't been introduced to the definition so this is sort of an essay question]. Let [itex]\deg(p(t)) = d_p[/itex] and [itex]\deg(q(t)) = d_q[/itex]. Let [itex]P_{min}(x,y)[/itex] be a polynomial of minimal degree among those defining an algebraic curve that contains the image of f. What can you deduce about the degree of [itex]P_{min}[/itex]?
     
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