@nbro Well you need to define it as a measurable function $(X,\mathcal{A},\mathbb{P})\to(\Bbb R,\mathcal{B}(\Bbb R))$ where the first is a probability space and the second uses the Borel $\sigma$-algebra if you want to be precise, but those are concepts most students won't be familiar with when taking a first course in probability
Prove that $${^{404}\mathrm C_4}-{^4\mathrm C_1}\cdot{^{303}\mathrm C_4}+{^4\mathrm C_2}\cdot{^{202}\mathrm C_4}-{^4\mathrm C_3}\cdot{^{101}\mathrm C_4} =(101)^4$$
I tried writing $101=102-1$, but couldn't move forward.
Sorry for the inconvenience, I am new here and couldn't type it.
'.' Me...
Sorry, if this is too basic a question. But I'm confused: https://math.stackexchange.com/questions/2676401/can-we-think-of-k-as-a-linear-operator?noredirect=1&lq=1
@MoreAnonymous I don't really understand what is going on there. You have two specific matrices, $A_0$ and $A_1$ (though they depend on a choice of a small enough $s$ to make everything converge). Defining $\widehat{K}A_0 = A_1$ doesn't say anything about what $\widehat{K}$ should do to any other matrix, so it makes no sense to ask if it is linear.
@ÍgjøgnumMeg "so ignore him" Really an excellent piece of advice, to "ignore" mathematical objections made at one's own post, simply by inventing a convenient "rudeness".
@Did Perhaps I should've been more tactful with the wording, but I don't appreciate the un-pedagogical "tone" some people take when commenting on this website so my usual reaction is to take onboard the mathematical content and ignore the wording :)
@ÍgjøgnumMeg Sorry but you explicitely recommended to ignore the maths in TSF's comment, for (dubious) considerations of rudeness. Doing so, you aligned with a strong, but most unfortunate, tendency of the site. In the end, TSF did make a mathematical point, right? If the point is sound, it should be addressed, for the OP's own sake (if it is not, one may explain why it is not and pass to something else).
Proposition 7.7.15: Let $\frak{a}$ be a two-sided ideal in the ring $A$. Then an $A$-module $M$ admits an $A/\frak{a}$-module structure compatible with the given $A$-module structure if and only if $\frak{a} \subseteq \mbox{Ann}(M)$. Such an $A/\frak{a}$-module structure is unique if it exist...
I'd also like to grade one of the exams and see how far off my grading is from hers. With my almost 40 years of experience teaching US students, I mean. :)
@TedShifrin One of the reasons I found the exam questions a little more reasonable than you might have is: two of the more nontrivial problems were from inclass notes, and the homework exercises were a lot harder than these exam problems.
@TobiasKildetoft I have a sequence and a method to go from sequence $1$ to sequence $2$. It follows properties such as $hat K (S_1 + S_2) = \hat K S_1 + \hat K S_2$
I was afk after I posted this ... Can anyone help? https://math.stackexchange.com/questions/2676401/can-we-think-of-k-as-a-linear-operator?noredirect=1&lq=1
sorry if this is stupid question, but if we have $\frac { 10*{ log }_{ 7 }10 }{ log_{ 7 }abc...j } =10!$, can we simplify to $\frac { 10*10 }{ abc...j } =10!$
No ... I have two different email accounts. I can't send uga email through the uga (Microsoft) server, so I have a dummy gmail account for sending with my uga address. It used to work fine. But it's messed up.
I know. But I was suggesting analyzing by fixing the two 2-planes and varying the vectors, then varying the two $2$-planes. Of course, you can always rotate to make one of the $2$-planes the $xy$-plane, for example.
@Semiclassic: You ignored what I said ages ago. This is why I don't find this fruitful. But you do need to figure out what condition on $M$ will result in unit columns. I said that ages ago.
I guess $M = \begin{bmatrix} \cos\alpha & \cos\beta \\ \sin\alpha & \sin\beta\end{bmatrix}$ for $\alpha\ne\beta$.
I conceded that point in my original remarks yesterday, @Semiclassic. But I was suggesting studying how the function changes when you change the basis vectors. I guess that amounts to using $M=X^{-1}Y$ where $X,Y$ are as above.
I suggested using it as a jumping-off spot and then seeing how the function varies when we vary the bases, which amounts to using what you just made me write down.
That is sort of interesting, though. I wonder if one can characterize maps that change one unit basis to another.
@DarkRunner: This has nothing to do with what you were writing.
Prove that $${^{404}\mathrm C_4}-{^4\mathrm C_1}\cdot{^{303}\mathrm C_4}+{^4\mathrm C_2}\cdot{^{202}\mathrm C_4}-{^4\mathrm C_3}\cdot{^{101}\mathrm C_4} =(101)^4$$
I tried writing $101=102-1$, but couldn't move forward.
Sorry for the inconvenience, I am new here and couldn't type it.
'.' Me...
Proposition 7.7.15: Let $\frak{a}$ be a two-sided ideal in the ring $A$. Then an $A$-module $M$ admits an $A/\frak{a}$-module structure compatible with the given $A$-module structure if and only if $\frak{a} \subseteq \mbox{Ann}(M)$. Such an $A/\frak{a}$-module structure is unique if it exist...
![[They do not move.]](https://imgs.xkcd.com/comics/interaction.png "[They do not move.]")
