linear algebra questions

Dear Dr. Kim
I have a few small questions I would like help with.

1) On examination paper 2003 Q5 a) we are asked to find the HCF of (x^4 +1) and (x^3 +1). Then find polynomials p,q s.t. pf + qg =1

I have tried this question a few time and I keep on getting the HCF to be (x-1) which means I cannot do the second part of the question. Can you see what’s wrong here?

2) Theorem 2.2.31 was not gone through in class, will we be expected to be able to reproduce this?

3) In the proof of Theorem 3.2.47 we take a factor of M^-1 out of our equation, I don’t see how this is valid.

4) On page 52 of the notes 5.4.136 there is a notation A is an element of O_n(R). What does O_n mean?

5) Towards the end of the notes some of the theorems mention a Euclidian plane, there is no definition of this, do we have to know what it is?

Reply:

(1) If you get this wrong, I suspect you might be misunderstanding the Euclidean algorithm. Here is the kind of computation you should be doing:

(x^4+1,X^3+1)=(-x+1, x^3+1)=(2)=(1)

alternatively, you might need to check your work for simple computational errors in the division algorithm for polynomials.

(2) The ability to use the fact correctly is of course essential to doing problems like (1). The proof of this theorem, which is easy, was not discussed in class because it’s more important to be able to divide efficiently than to give a proof that it can be done. But, as with all assertions we’ve gone over, it would be good if you had a good sense for why the statement is true. The weight is never on `reproducing’ proofs, as I’ve emphasized many times.

(3) I don’t quite see your objection to the equation. Perhaps you didn’t notice that there is also an `M’ on the right side of the expression? Ask me again if you can’t figure it out after further scrutiny.

(4) O_n(R) refers to the group of real orthogonal matrices. Check the archive of past linear algebra posts on this blog where this question was answered already.

(5) Yes, you should know what the Euclidean plane is. It is R^2.

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