hi, i had a question about determinants. i currently find them very unmotivated. of course there is the notion of "signed volume" and the determinant measuring how this volume changes under a linear map. but i find this a bit unsatisfying, firstly because even in the real case if one is not doing e.g. integration i'm not sure why i should be thinking about volumes, and secondly because this definition works for any vector space on any field (so no measure is required)
the other definitions i know of having to do with exterior powers are just as problematic since i don't really understand why one would look at those either
hopefully i'm not coming off as impossible to satisfy!
everyone has different notions of what constitutes motivation. if you look on that page the majority seem to like the geometrical approach.
people who just like doing stuff with algebra and weird fields and things might like just focusing on alternating multilinear maps. you don't need to introduce exterior algebras to deal with them. but many find this approach too dry
yeah! i have been understanding it using the map on the top exterior power. and i agree the geometrical approach is nice, sorry for the subjectivity of the question. maybe this is just a matter of getting used to alternating multilinear maps, but since i am not so experienced with a lot of math i have little experience with even the relatively common ones (e.g. commutators)
so i still find them unmotivated, but it might be one of those "motivated by its consequences" things
if it makes you feel any better, with many classes of operators on infinite dimensional spaces it is possible to prove the nonexistence of anything like a determinant. pick whatever properties you want to extend, they don't go through.
i think one route to intuition is, well, historically it came out of geometry and solving systems of equations, and once you have it, you end up mostly working with through its properties because the formula sucks. and then why not step back and just consider those properties.
like so many other things in math, it may have emerged out of the festering swamp called g--m-try.
i think one other thing which makes me curious is the fact that we can speak of determinants and their properties without reference to any choice of basis, purely as properties of linear maps, but this might be more a consequence of the preference for abstraction than anything
the top exterior power is pretty much just the "vector space of oriented n-dimensional parallelipipeds in V" (where n=dim(V)). this is a one-dimensional vector space and a linear map f:V->V induces a map on this exterior power, which, by one-dimensionality, is necessarily given by multiplication with a scalar. this scalar is the determinant of f, i.e. the determinant is the factor by which the linear map scales parallelipipeds of the top dimension.
this is still geometric, but perhaps it makes it clearer that this does not really require a notion of "volume" or "measure"
definitely i like this answer/motivation, as the multilinear alternating map definition is an interesting algebraic characterization of the determinant (and is beautifully motivated by signed volumes), but maybe my question is more, "why care about multilinear alternating maps especially if we're not doing volumes?", or "why care about exterior powers?"
of course, i know one can, to the annoyance of people trying to help, always play this "why care about XYZ" game, and i'm not trying to do that! i was just wondering if we could go "one step further" than alternating multilinear maps.
of course personally i find linear algebra interesting and beautiful, which is more than enough reason to study it :)
yeah, this is an interesting point. maybe there is some "natural" condition we could impose on a homomorphism $\text{End}(V)\rightarrow k$, $k$ the underlying field, which spits out only the determinant? this seems related to the invertibility property
oh, actually, would any such homomorphism preserve invertibility?
never mind that's not true
or maybe it is but the proof i had in mind doesn't work i think
Hi, if $\frac{1}{2}t_2\leq t_1\leq\frac{1}{2}$, how can we find the bounds for $\frac{2t_1-t_2}{1+t_2}$, i.e. what would $x,y$ be in $x\leq\frac{2t_1-t_2}{1+t_2}\leq y$? Thank you!
An $m\times n$ row reduced echelon matrix (rref) $A$ can also be described as either all entries of $A$ are zero or there exists a natural number $r$ ($1\le r\le m$) and $r$ natural numbers $c_1,c_2,\cdots, c_r$ such that 1) $c_1\lt c_2\lt ...\lt c_r$, 2) $a_{ij}=0$ for $i>r$ and $a_{ij}=0$ for $j\lt c_i$, 3)$a_{ic_i}=\delta_{ij}, 1\le i\le r, 1\le j\le r$.
$\delta$ represents Kronecker's delta. I don't understand (3).
The problem with $(3)$ is: $a_{1c_1}=\delta_{1j}$ then $j$ can vary here hence we have on RHS a set of values but LHS is only one value.
but beware that in the wild, sometimes "row reduce" only means "put in row echelon form" i.e. zeros to the left of pivots but not necessarily above and below them.
i looked in my lecture notes to see if i had a good slide on this, because i want to compete with ted in promoting my own offerings.
An $m\times n$ row reduced echelon matrix (rref) $A$ can also be described as either all entries of $A$ are zero or there exists a natural number $r$ ($1\le r\le m$) and $r$ natural numbers $c_1,c_2,\cdots, c_r$ such that 1) $c_1\lt c_2\lt ...\lt c_r$, 2) $a_{ij}=0$ for $i>r$ and $a_{ij}=0$ for $j\lt c_i$, 3)$a_{ic_i}=\delta_{ij}, 1\le i\le r, 1\le j\le r$.
But still, I am having a hard time digesting how can a singleton value equal a set
so my notes spend several slides on row echelon form, with blanks where we would have done examples. then there's one slide saying what RREF is, two blanks for two examples, and then the slide says "The book calls this 'Gauss-Jordan elimination.' You can use it, or not. I will not separately test this topic and if you can put a matrix in row echelon form and solve systems of linear equations without it that is fine with me."
my attitude was, the arithmetic mistakes you do with backsubstitution are really no different from what you do with matrix operations, so i'm not going to ask you to go to RREF unless you want.
koro one thing to keep in mind is that people tend not to formally reason with matrices in RREF, so it is not really that helpful to symbolically express what RREF is. it's a calculational tool.
there's the fun fact that the product of RREF matrices is RREF, but who RREF'in cares, as far as i'm concerned.
he takes one journal and gives examples of how its typesetting evolved over the years, pointing out the good and the bad. very fun reading if you're into it.
also, tex hadn't fully matured yet, so there's a fun work-in-progress flavor to it
copper i don't know if you saw it, but when i taught linear algebra it got one slide that ended in "you can use it, or not."
"shew" was hopefully gone by 1950. unless people were putting on an affectation.
the US patent office is horrible at typesetting math. they will fail to subscript and superscript, they will miss fraction bars, everything. usually because the attorneys submitting the work regard all of it as noise anyway and nobody reads it.
In Spanier AT (p.325), there is a notion $\Gamma(\mathcal{U})$ called the module of compatible $\mathcal{U}$ families of $\Gamma$ where $\mathcal{U} = \{U\}$ is a collection of open sets and $\Gamma$ is a presheaf of modules on $X$.
Definition. A compatible $\mathcal{U}$ family of $\Gamma$ is an...
did we lose the cold war? are people still saying genosse?
one time i was in stockholm and i was with some extreme left-wing friends, and a taxicab driver refused to drive us to the bar we wanted to go to. i didn't know at the time, but it was some kind of hotbed of swedish communsim.
probably in my FBI file now.
we were going there for the pricing of the refreshments, not the politics. i didn't plan on reading any marx while i was there.
my main memory was people were not interactive and alcohol was prohibitive. returned quickly to copenhagen on the ferry where alcohol was less prohibitive.
i met someone at a party once for about 2 hours, and dumped like half of my life's story on her. we're friends on linkedin and still check in from time to time.
she knows stuff about me that my wife doesn't know.
and i know an enormous amount of stuff about her life that her husband doesn't know. this is continents apart so there is nothing pervy about any of it.
it's just easier to talk to people if you know there won't be some kind of immediate interruption in your social circle.
Hello. I have a question. So in a game, if you win in the $i'th$ round you are paid $2^i$ dollars and to play the $i'th$ round you have to pay $2^{i-1}$. The probability of you winning in any round is 0.5. Whats the expected loss player incurs before profiting from the game. So if I win in round $i$ my win trumps all my losses, so I made a profit.
So let X denote the money lost in round $i$ (such that I have never won before) and p(X) be its probability. So the expected money lost upto round $i$ would be $\frac{2^{i}-1}{2^{i}}$ and I took its limit to infinty. Am i correct?
@Shobhit Beautiful problem... i dont have the mathematical skill required to solve it so I am going to brute force it.
So if you win the first round you get 2^1=2$ and loss 2^(1-1)=2^0=1$? So if you win 1st round, your win trumps all your losses and so you made a profit right?
His question was "What's the expected loss player incurs before profiting from the game" and we both figured out profit occurs from first-round itself right? so the answer to the "expected loss player incurs before profiting from the game" is 1$ right?
Oh probability is 0.5 I havent looked at that part yet while brute forcing.
Winning on the first round is probability $\frac12$ with a cost of $1$. winning on the second round has a probability of $\frac14$ with a cost of $3$. winning on the third round has a probability of $\frac18$ with a cost of $7$. That is the probability of winning on round $n$ is $2^{-n}$ with a cost of $2^n-1$. Adding up the losses times their probabilities gives a divergent series.
At the $QR$ decomposition with permutation matrix is the matrix $R$ equalt to $R=G_3^{-1}P_1G_2^{-1}P_0G_1^{-1}A$ or $G_3P_1G_2P_0G_1A=R$ ?
Which is the correct one?
In general it holds that $QR=PA$, right?
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as <, this is stated in formal logic as:
∀
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X
∀
y
∈
X
(
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x
<
y
∧
¬
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y
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In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan's laws.
The law is also known as the law (or principle) of the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur: "no third...
Lets say we have x points with us on a line. The cost of visiting any point is 1. After we visit any point once, it stays visited forever. Each time, we randomly visit any point on the line which has not yet been visited. We keep doing this until our points become such that out of any n consecutive points, more than one has been visited. What is the expected cost in order to achieve this state ?
Could anyone help me with this ? Can't even figure how to start :/
@geocalc33 just for clarity, would you explicitly write out the terms? it looks as if you might have intended $\sum\limits_{k=0}^\infty\frac{(-1)^k}{k!}\zeta(-ks)$
Hi. I am studying PCE (Polynomial Chaos Expansion). If anybody has knowledge about this and wants to have a discussion, feel free to ping me.
This discussion can be in any form which makes the other people comfortable. (Starting from a voice call where the opponent doesnt need to talk for privacy issues but I can explain the thing and you can see holes in my understanding or ask me questions about it, to some text messages in the format mentioned above.)
In your answer, @robjohn , you stress recognizing the independent variable when confronted with the little-o notation. In this particular case, you recognize both $h$ and $u$ as independent variables, correct?
@schn Can you look at the definition of $o(g(x))$ and try applying it to a given situation instead of trying to abstract a calculus of little-o? It applies differently in situations where the estimate needs to be uniform in another variable from where it doesn't. It is highly context dependent, and because of the definition, it is not easy to specify uniform little-o-ness.
@geocalc33 just for clarity, would you explicitly write out the terms? it looks as if you might have intended $\sum\limits_{k=0}^\infty\frac{(-1)^k}{k!}\zeta(-ks)$
@geocalc33: there may be no zeroes between the poles. They are order one, but the signs of the residues alternate, so it could look like cosecant where it never crosses the axis.
Let's say $S, L$ are two nonproperly embedded noncompact submanifolds (without boundary!) of a manifold $M$, and $S$ is contained in the frontier (or topological boundary) of $L$, $S \subset \partial L$.
I take a tubular neighborhood with shrinking width $\nu(S)$, $\nu(L)$ for both $S$ and $L$, respectively, let's say with tubular projections $\pi_S$ and $\pi_L$ respectively.
I would like to choose a smaller tubular neighborhood with shrinking width, $\nu'(S) \subset \nu(S)$, such that $\nu'(S) \cap \nu(L) \subset \nu(S) \cap \pi_L^{-1}\nu(S)$.
($\pi_L^{-1} \nu(S)$, of course, means $\pi_L^{-1}(\nu(S) \cap L)$)
Here is the point. $\nu(S) \cap \nu(L)$ is wherever the pair of tubes intersect. But $\nu(S) \cap \pi_L^{-1} \nu(S)$ is a subset of $\nu(S) \cap \nu(L)$ consisting of points $p$ such that $\pi_L(p) \in \nu(S)$ as well.
Apriori this can be a strict subset. Not all points in the intersection of the pair of tubes need to map in $\nu(S)$.
@geocalc33 It appears that $f(0)=-\frac1{2e}$ and $f'(0)=-\frac1{2e}\log(2\pi)$, which doesn't comport with what I said earlier. So one or the other approach is not correct. I will get back to this later. Sorry.
So you're not talking about shrinking radii of normal disks. In the case of a point (as in your example), you have to take a weird open set, not a tiny disk.
It depends on how $\pi_L$ behaves. The picture illustrates an example of $\pi_L$ by drawing its level curves. In this case note that $\pi_L(\nu(S) \cap \nu(L)) \subset \nu(S)$.
But imagine that the level curves were arranged a little differently, then this might fail.
The number of primes is infinite. This is clear. Since each prime number is a natural number and the number of natural numbers is $\aleph_0$, the number of primes is $\aleph_0$. However, I have heard people state that "infinity is not a quantity"; I don't get it. Their main objection is that infinity would be "quantifying the unquantifiable" - an absurdity. I tried asking about this on MSE but the question got shot down. What are your thoughts on this, folks?
@Koro To solve $a_{n+1}=\frac 1{4-3a_n}$ using that solution multiply by $\sqrt3$ and get $\sqrt3\,a_{n+1}=\frac 1{\frac4{\sqrt3}-\sqrt3\,a_n}$ then set $b_n=\sqrt3\,a_n$.
Riemann Hypothesis is equivalent to the integral equation
$$\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt =0$$
Many other integral equations exist that are equivalent.
How to show that they are equivalent ?
They usually include absolute value of a function.
Why is that...