In my Linear Algebra book, a linear transformation $F$ is defined to be a transformation that satisfies the equality $$F(s\mathbf{u}+t\mathbf{v})=sF(\mathbf{u})+tF(\mathbf{v}),$$ where $\mathbf{u}$ and $\mathbf{v}$ denote vectors, and $s$ and $t$ scalars. Now, what about $$G(\mathbf{x})=\mathbf{a} +b \mathbf{x},$$ where $\mathbf{a}$ is a constant vector and $b$ a fixed scalar. It seems like this also should be a linear transformation although according to the definition it isn't.
We know on teh reals that only ae^t is the derivative of itself, is there an analogous statement, for higher dimensions of R where functions equal to their derivative imply they are exponential type, or maybe other fields, like C
In contexts where I understand things well enough to have an opinion about your question, the exponential function is defined in such a way that it is its own derivative.
A common definition of $\exp$ is that it is the unique function which (a) satisfies $\exp' = \exp$ and (b) $\exp(0) = 1$. This works pretty well in $\mathbb{R}$ and $\mathbb{C}$.
The matrix exponential is typically defined via a power series. So, for example, if $A$ is a matrix, then we might define $$\exp(A) = \sum_{n=0}^{\infty} \frac{A^n}{n!}, $$ subject to appropriate hypotheses.
A similar definition works for linear operators in general (though, again, I am eliding details).
One can formally define the derivative of a power series as $$ \frac{\mathrm{d}}{\mathrm{d}x} \sum_{n=0}^{\infty} a_n x^n = \sum_{n=1}^{\infty} a_{n} n x^{n-1}. $$ With respect to this definition, the matrix exponential satisfies $\exp' = \exp$.
My recollection is that the $p$-adic exponential is also defined via a power series expansion, but this is interesting, as that power series does not converge outside of a little disk around zero. But differentiation on the $p$-adics is weird, so *shrugs*.
@BalarkaSen What does thinking like an analyst mean? Analysis can be formal and algebraic, so I'm wondering what I'm missing that you're probably referring to
yeah, the $p$-adic exponential is also defined that way
though i dont have the slightest clue whether thats a unique characterization in that case
i think differential equations over the $p$-adics are horribly behaved, since derivative $0$ doesnt imply constant anymore due to total disconnectedness or something along those lines
but some soundbites from analysis that haunt me in my everyday life are either elliptic PDEs, Sobolev spaces, index theory, etc or stochastic calculus, Gaussian processes, etc
if that gives an idea of what im thinking of when i say analysis
if i see $x < y$ i can chant that in my head and convince myself i understand it: "$x < y$, $x < y$, $x < y$, ..." and then an hour later i'd be chanting "$x > y$, $x > y$, ... or wait, what?"
when i was an undergrad, the computer science major had a limited enrollment. you had to apply to it and not everybody got in. cognitive science was a popular fallback because you could use a lot of comp sci courses for the major and you could still say you were majoring in "CS."
i'm mostly joking but i did know one or two people who would not say no if asked if they were CS majors, even though it was obvious the asker didn't mean cognitive science.
another popular fallback was the math major, which was kind of ironic, because often the reason people didn't get into the CS major was their grades in math. "aww crap, i had to take calc 2 twice. i guess i have to major in .... math."
that was sometimes an adjustment for grad student TAs who had gone to undergrad at schools without a big engineering/CS presence, or where a majority of math majors actually specifically chose math.
Suppose we define $A$ to be finite if and only if $A$ is empty or $|A| = \{1,...,n\}$ for some $n\in\mathbb N$. Then we define $A$ to be infinite when it is not finite. The theorem "$A$ is infinite if and only if there exists a $B\subsetneq A$ such and a surjection $B\to A$": does it need AC?
I feel like $\implies$ direction needs AC to construct a surjection $f\colon A\to\mathbb N$.
that could be. have you checked a set theory book that cares about stuff at this level? enderton might have it.
lots of stuff about 'comparing' infinite sizes ends up hinging on AC (or some other axiom) here and there, at least for some directions of things that feel like they ought to be equivalences.
i understand i could have answered that question the semester i took a set theory class. i deleted that part of my brain afterwards.
one tihng you could do would be to post on MSE, title "set theory question," and begin "ok, so i have an urn with 120 red balls and 60 blue balls, and if i draw 10 balls without replacement --- haha, just kidding, asaf, but now that i have your attention, [your question]." and tag it "set-theory"
that's where i began to tune out. enderton treats things as "with AC" or "without AC" and doesn't get into all of the various choice-like principles that form a gradient of things between the two.
thorgott: i don't either, but there definitely are things in e.g. cardinal arithmetic that seem like they shouldn't need one to pick a flavor of choice, but somehow do. this maybe isn't a good example of that.
@Thorgott @anak There exist models of ZF having an infinite Dedekind-finite set. Let A be such a set, and let B be the set of finite injective sequences from A. Since A is infinite, the function "drop the last element" from B to itself is surjective but not injective, so B is dually Dedekind-infinite.
However, since A is Dedekind-finite, then so is B (if B had a countably infinite subset, then using the fact that the elements of B are injective sequences, one could exhibit a countably infinite subset of A).
copied from wiki, i dunno what this means
A is dually dedekind infinite = there is a function A -> A which is surjective but not injective
infinite Dedekind-finite dually Dedekind-infinite is my rapper name
