A bit of MATLAB, a bit of linear algebra

[Kouheikun]

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!

 

[LeBrad]

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?

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

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.

 

[Kouheikun]

The dot product! And if the dot product is zero then the vectors are perpendicular.

Awesome thanks!