Recent content by xalvyn

\left| ad - bc \right|=1, but ad - bc may not necessarily be 1. For example, the columns of \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} are orthonormal, but det \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}=-1. I guess it would be easier to work from the relation \begin{pmatrix} a & b \\ c &...

I think you have to consider the cases b>1 and b<-1 separately. (1) When b>1 : The pole at z=-b- \sqrt{b^2-1}<-1 lies outside the unit circle, while the pole at z=-b+ \sqrt{b^2-1}= \frac{1}{-b- \sqrt{b^2-1}}, since \left| \frac{1}{-b-\sqrt{b^2-1}} \right|<1, is enclosed by the unit...

Try \prod_{k \neq i} (x-x_k), a polynomial of degree n - 1, and divide this expression by an appropriate constant (try a product of factors). For more information, look up: Lagrange interpolation polynomials.

Hi all, is there a general way of proving that sqrt(r1) + sqrt(r2) + sqrt(r3) + ... + sqrt(rn) is irrational, given that none of r1, r2, r3, ..., rn is the square of a rational number? (or is this statement even true in general?) for the case when n = 2, the proof is quite...

ok i think i understand now, thanks :)

Hi, the answer sounds obvious enough now :) however, i was wondering if there is a rigorous way of proving this very elementary and trivial result. for if we assume that if p1/q1 < p2/q2, then sqrt(p1/q1) < sqrt(p2/q2), we'll have to prove this using 'dedekind section arguments'...

Hi all, I think this sounds like a really simple and trivial question, but I've no clue as to where i should start: true or false? between any two different positive rational numbers lies the square of a rational number. while i can't provide a construction of such a number, i somehow...

hi all, have a rather sticky problem on determinants to deal with...hope someone can offer some help. [In what follows, the numbers in brackets denote suffixes, so that, for example, A(s)(j) refers to the element in the sth row of A and jth column of A.] Let A be an n x n matrix. Let s be a...

i meant it like this: Since every row of A can be expressed as a linear combination of the non-zero rows of E, and every row of A'A can be expressed as a linear combination of the rows of A, every row of A'A can also be expressed as a linear combination of the non-zero rows of E. By a...

hi... was wondering, does the converse of the mean value theorem hold? that is, given any function f(x), and a tangent to the graph of y = f(x) at any point, can we always construct two points on the graph (with the tangent lying between) such that the line joining them is parallel to the...

hi...been quite a while since I last posted. To 0rthodontist...hope I got your hint right. As I've only just started studying orthogonality, I'm not totally familiar with the concept. Here's how I reasoned: 1) Since null space of A' = space of vectors orthogonal to column space of A, AX...

hi Thanks for taking time off to look at my question. I'll just try to fill in the gaps of my original proof and correct the errors: 1) E is not necessarily independent. Yes, it was a misstatement - i actually meant to say that the non-zero rows of E are independent, unless every row of E is...

ok, it seems that I've answered the question myself, but I'm still not quite satisfied with the solution. My proof that null space of A = null space of A'A runs roughly as follows: 1) Suppose we have reduced A to its reduced row echelon form. Call this reduced row echelon form matrix E...

Hihi I've been working on this problem for some time: if A is a (m x n) matrix, and A' denotes its transpose, then the null space of A is equal to the null space of A'A. Is this always true? I thought of a proof for the special case where A is given in reduced row echelon form, but fail to...

Hi StatusX, Thanks a many. While reading your reply, I was thinking: is it possible to define this pseudo-scalar product r # X = 0, where X is a vector in R^N, and so forms an abelian group under addition ? In this case, we have 2a) r # (X + Y) = r # (Z) = 0 = 0 + 0 = r # X + r # Y, where...