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A bit of MATLAB, a bit of linear algebra

  1. Feb 6, 2006 #1
    I'm taking a course on numerical methods using MATLAB, and right now we're discussing linear systems. This is the question I was given.

    P(x) = (3/2)x^2 - 1/2
    Q(x) = (5/2)x^3 - (3/2)x

    Compute vectors P and Q using the following MATLAB commands.

    X = -1:1/500:1;
    N = length(X);
    P = p(1)*X.^2 + p(2)*X + p(3)*1;
    Q = q(1)*X.^3 + q(2)*X.^2 + q(3)*X + q(4);

    Where p and q are the coefficient vectors of the polynomials P(x) and Q(x).

    Compute and give the numerical result for:

    (The sum from i=1 to N) of (Pi Pi)
    (The sum from i=1 to N) of (Pi Qi)
    (The sum from i=1 to N) of (Qi Qi)

    What linear algebra operation is this? Express your answer in vector notation.

    Based on the results of the above summations, what can we infer about P and Q and why?


    I'm fine all the way up to the linear algebra questions.

    I understand that X is a vector containing 1001 evenly spaced numbers between -1 and 1, and so vectors P and Q are vectors containing 1001 values of the functions P(x) and Q(x) respectively within the range x=-1 to x=1.

    As for the summations, I'm using sum(P.*P) and the like to get these three sums (in order):
    201.0020
    -2.4425e-015
    143.8611

    So what linear algebra operation is this? I first thought of integration, since it's a summation of a function across a range of x values, but that's not linear algebra. I don't ever recall individual squaring the entries of a vector in first year lin alg. The statement "express your answer in vector notation" suggests that the three sums make up a notable 3x1 vector, but again, I'm completely at a loss.

    Did I do the summation wrong, or do I just need to brush up on linear algebra? Help!
     
  2. jcsd
  3. Feb 6, 2006 #2
    If P and Q were 3 dimensional vectors living in x-y-z space, such that P = [Px Py Pz] and Q = [Qx Qy Qz], then sum(Pi*Qi) = PxQx + PyQy + PzQz

    Does that look like any linear algebra operation that you know?

    This says that P is not zero and Q is not zero, but this operation on P and Q is zero. When you figure out the operation, the fact that sum(Pi*Qi) gives zero should tell you something about those two vectors.
     
  4. Feb 6, 2006 #3
    The dot product! And if the dot product is zero then the vectors are perpendicular.

    Awesome thanks!
     
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