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Jul
10
reviewed Approve sequences-and-series tag wiki excerpt
Jul
10
comment A question on Green's functions & integral operators
This is a fantastic question, answer and use of the MSE. Bravo. That said, you might also get something from my much less enlightened calculations at: supermath.info/TransferFunctions.pdf where I derived the integral kernel from other methods for a few standard examples and connect with some of the engineering jargon about transfer functions. Understanding Green's function better is something that's been on my to-do list for a while...
Jul
9
comment Indices on Commuting Matrices
$A$ is a matrix. $A_{ij}$ is a number ( or whatever object make up your matrices). So if the objects in the matrices commute then $A_{ij}B_{kl} = B_{kl}A_{ij}$. This is part of the reason for using component notation, it allows the usual arithmetic of the objects. In contrast, $AB$ is shorthand for a rather complicated combination of elements of $A$ and $B$ of which there is not particular reason to expect $BA$ form the same pattern.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 ok now I see what you are saying... I just wanted the question to be more interesting than the notational question... I asked an expert on this stuff and he tends to doubt the validity of my proof as well just on the principle of the thing. But, he hasn't looked at it yet.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 wait a minute, the whole reason I tried to embedd $A$ into $n \times n$ matrices was your comment yesterday, "Unless you find a way to embed A into Mn(k) this answer has no reason to exist." which now that I read the original question again, I probably should have ignored. If I read the original problem correctly, we just needed to embedd $A$ in the $(n+1)\times (n+1)$ matrices. So, the difficult second half of my answer is not needed.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 perhaps you can improve by adding your own answer? I do grant that my current write-up has not explicitly addressed the issue of injectivity and it would be interesting to see how this all fleshes out in your Abelian group with the trivial multiplication.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@man_in_green_shirt I mean in particular $A_M'$ is an $(n+1)$-dimensional subspace of the $(n+1) \times (n+1)$ matrices.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 ok, that explains the problem. We have injectivity because $A'$ has the multiplicative identity.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 I use it, so, apparently "never" in incorrect ;) Thanks for looking at it later.
Jul
9
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
Well, that cost me longer than I had hoped, I bought the book, maybe I'll have more to say once I see it...
Jul
9
revised Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
details on how to steal n from n+1
Jul
9
comment Finding differential equation from the solution.
Note $y'=2ax+b$ is not a single ODE, but, a parametrized family. Admittedly, these are semantic distinctions I raise here.
Jul
9
comment Finding differential equation from the solution.
The answer is $y''=0$ because he is told the given family of functions is the "general solution". That means $y=Ax^2+Bx+C$ is not just a solution for token $A,B,C$, but, also includes all possible solutions to the mystery ODE.
Jul
8
reviewed Reject $\lim\limits_{n\to\infty} \sup \sqrt[n]{n^k}$ for $k \in \mathbb{N}$?
Jul
8
comment Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$
@user26857 that is true, it's not a complete answer just yet. I'd like to hear back from the OP before I work on it further, but I'll probably try to hack through it later today.
Jul
8
reviewed Approve Equivalent Definition of Measurable set
Jul
8
reviewed Approve How to reach $\dfrac{(n-1)n(2n-1)}{6n^3}$
Jul
8
comment Implicit function theorem conclusion notation?
In short, to prove the theorem, you linearize the level mapping then focus on the particular submatrix to eliminate. Invertibility of the submatrix is captured by the Jacobian determinants. Don't be bothered by this, it's not magic, it's just the usual linear algebra and you can see the determinants as stemming from Cramer's Rule. I don't have that in my answer, but I hope there is enough to "see" it...
Jul
8
answered Implicit function theorem conclusion notation?
Jul
7
comment Parametrization of a line segment
Maybe it wants you to use sine and cosine ? Or, $\tan \theta = y/x$ ? Also, maybe crop that picture to remove your name :)