Recent content by Hawkeye18

You need the following simple fact/lemma: Let a function ##f:X\to Y## be fixed, and let ##\Sigma## be a sigma-algebra on ##X##, and ##\mathcal C## be some collection of subsets in ##Y##. Assume that for any ##A\in\mathcal C## the inverse image ##f^{-1}(A)## belongs to ##\Sigma##. Then...

Just my 2 cents. First, about importance of the determinat: Determinants are extremely important in mathematics. For a real matrix, the determinant is the oriented volume of its colums (rows); I do not think anybody would deny importance of volume and orientation. Because the determinant is...

The series ##\sum_{n=2}^\infty (n(n-1))^{-1} = \sum_{n=2}^\infty \left( \frac{1}{n-1} - \frac1n \right)## does sum to 1 because ##\sum_{n=2}^N \left( \frac{1}{n-1} - \frac1n \right) =1-1/N##

Hi Leo-physics, if you know how to prove this for a rotation in ##\mathbb R^2##, then you can get the desired statement from results in s.5 of Chapter 6 of "Linear Algebra Done Wrong", see Theorems 5.1, 5.2 there.

I think your teacher meant the definition of a matrix as a function with domain being the set of pairs of integers ##(j,k): 1\le j \le m, 1\le k\le n##. In this definition the value of this function at a point ##(j,k)## is ##a_{j,k}##. That is a general point of view in abstract mathematics...

I used "rotation" in quotes for the lack of better term, it is not always a real rotation. Your example is up to similarity what I call "rotation" see post # 8 above for the details.

Let me elaborate. In the class of real matrices, all the solutions are given by ##A=SRS^{-1}##, where ##R## is a "rotation" (note the quotes, that is not always a real rotation), and ##S## is invertible. "Rotation" here mean that ##R## is the block diagonal matrices, where each block is either...

I believe you are talking here about a ##2\times 2## matrix. Then your antidiagonal matrix is not a rotation matrix, but it is a matrix of form ##SDS^{-1}##, where $$D=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).$$

In the class of complex matrices all the solutions of the equation ##A^n=I## are given by ##A=SDS^{-1}##, where ##S## is an invertible matrix, and ##D## is a diagonal matrix whose diagonal entries are ##n##th roots of unity, ##d_{j}^n=1##. In the class of real matrices all the solutions are...

The difference between holomorphic and analytic functions is explained here: https://en.wikipedia.org/wiki/Holomorphic_function In short, as it was already mentioned above, analytic means that in a neighborhood of each point in its domain the function can be represented as the sum of convergent...

Here is the Lebesgue differentiation theorem, https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

Yes, definitely. The sets of zero measure can have quite complicated structure, they can consists of uncountably many points. For example, the classical Cantor 1/3 set has measure 0, and is uncountable. Some of it generalizes to any dimension ##n##. The Fourier transform of a compactly...

Yes, if an ##L^2## function is compactly supported, its Fourier transform is an entire function (i.e. analytic function on the whole complex plane), and thus it can have at most countably many isolated zeroes, accumulating to ##\infty##. If an ##L^2## function is supported on a half-line, then...

The method Erland described give raise to the following simple formula for the orthogonal projection: put vectors that form a basis in ##W## as columns in a matrix, call it ##A##. Then the matrix of the orthogonal projection ##P_W## is given by $$P_W= A(A^TA)^{-1}A^T. $$ To find the projection...

Notation ##d s^p## I used is pretty standard calculus notation used as a shorthand in change of variables, ##d s^p = d(s^p)=p s^{p-1} ds##, it is not a notation from the multivariable calc, just the standard notation for the differential in 1 variable calc. As for the change of variables, you...