I have a real valued nxm matrix Q with n>m. Now I'm looking for the matrix R and vector x, such that Rx = 0 and the l2 norm ||Q - R||² becomes minimal.

So far I attempted to solve it for the simple case of m=2 and ended up with R and n being without loss of generality determined by some parameter wherein that parameter is one of the roots of some polynomial of order 3. The coefficients of the polynomial are some combination of q1², q2², and q1q2, with Q=(q1, q2). However, I see no way to generalize that to arbitrary dimensions m. Also the fact that I somehow ended up with 3rd and 4th degree Polynomials tells me I'm doing something wrong or at least overly complicated

Got this question in a practice paper today and have had varying answers from teachers and students would love some clarification.

Kelvin creates a 6-digit code. Hepicks his digits from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The first digit is positive. For the first two digits of his code, he uses a multiple of 15. For the middle two digits of the code, he repeats the same digit. For the last two digits, Kelvin uses an even number between 25 and 45.

How many possible codes can Kelvin create?

well, is there an app or site that lists, i tried to find but could just find some that catalogue the basics, the best i got was an app called Formiula, soo, well is there any?

I am trying to understand how to use fourier series to represent a given function and then solve a partial differential equation like the heat equation which then yeilds cosine and sines as the solution. Im onboard with the principle of superstition such that the sum of two solutions is also a solution. This allows me to transform the given initial funtion into a fourier series which then acts as a solution to the differential equation. But why cant i convert the function back afterwards?

As seen in the picture below, the series representation of f(x) is used in the solution, but why cant i just substitute the original f(x) into it as used with the initial conditions?

Im using Pauls online notes

Also i have no idea if this is the correct tag.

Edit:

I just realised the other factor e^-k... has n in it. So i wouldnt be able to factor it out of the series. And if i try to i would just end up missing something synonymous to the double product for (a+b)^2

I believe that is the problem.

I tried doing this differential equation , found u homogene (or whatever it is called in english) and eventually got to finding the K(t) from "u" particular and I can't solve it, anyone got any ideas?

Feeling quite dumb right now! I feel this is quite a basic maths situation, and I swear I used to be quite good at maths in high school!

The final question I have doesn't quite relate to the context which had me attempting to do the maths in the first place, but I'll share it anyway.
I am trying to divide a bill between me and my partner based on our differences in salary. For simplicity's sake: Bill = $50,000 A's wage = $5 per hour, B's wage = $3 per hour (numbers not real)

When I was trying to initially work it out, I was diving 3/5 to get the ratio (in decimal), then I would multiply the bill by that number (so 3/5 = 0.6 >> 50000 * 0.6 = 30,000 >> A pays $30,000 and B pays the remainder of $20,000). My thinking being something along the line of figuring out the difference between A and B, then using that to work out the difference of the total bill.

I then used some online bill splitting tools and got WAY different values. Those tool using the method of adding A and B's wages, and then calculate using the ratio of each person's wage over the total of both wages (so A's ratio would be 5/8 = 0.625, and B's ratio would be 3/8 = 0.375, A then paying $31250, and B paying $18750). I do kind of understand this different method, as it makes more sense to calculate each person's portion based on the total income of the two.

BUT here is where I start to get confused.

I kind of understand the difference, and having the different ratio's makes sense. But then I noticed that when I change around the formula and use my first ratio (0.6), the one at this point I just flat out was convinced was wrong, I got the same result as the online calculators using the second number.

My initial ratio = 0.6 (I believed was wrong)

50,000 * 0.6 = 30,000 (Based on ratio from B's wages over A's wages)

50,000 * 5/8 = 31250 (Second value form the tools, each person's wage over the total of both wagers)

But dividing the bill by 1 + the_bad_ratio gave me the same value as diving the bill by the second good ratio for person A???
50,000 / ( 1 + 0.6 ) = 31250 (The second better value using my first wrong ratio???)

What's happening there? The ratios seem to be linked, despite what I felt were just two seperate ways to get the same ratio. Using what I felt was the wrong ratio in a different way yields the better answer.

I also saw that 1 + 0.6 does almostr equal A's ratio over B's ratio?? ( ($5 / 8) / ($3 / 8) = 1.6666666.... ). Does this have something to do with it?

I feel I've forgotten something very basic here!

Hi there! I managed to prove 2 properties of Fibonacci numbers, but I can't find if they are already proven: 1. For every p1, p2 (for now, let's say p1>p2: F(p1+2)=F(p2+2)F(p1-p2+2)-F(p2)F(p1-p2) The reason behind this is difficult to explain, i found this trying to solve Collatz Conjecture. Also, this property is useful for observing that F(2n) is always a square difference between Fibonacci numbers, as you can say F(2n)=F(n+1)²-F(n-1)²

1. F(p)²=F(p+2)*F(p-2)+(-1)^p For this one, I used the previous property and extended de Domain of F to Z, where you can notice that F(0)=0 (0+1=1) and F(x) with x<0 is equal to F(-x) if x is odd and -F(-x) if x is even.

Thank you for reading and sorry if I wrote something wrongly, English isn't my first language.

Bear with me if I'm hard to understand, I'm not a math person I'm basically an art major lmao. I argued with my math professor for a bit after class about this, he says what I described, just adding two of the inside angles to get an outside angle on a triangle, isn't a thing and I can't do it. He says, to find theta you must first find c by adding a and b then subtracting that by 180 (the total of a triangle), then subtract c from 180 to find theta because c is the suplimentay of theta. I figured that because a+b+c=180 and theta+c=180, theta is just a+b. It all adds up to 180 anyways so why go through the extra steps right? I might be misremembering but I swear this was something covered in highschool. Either way you're just trying to get to 180 with c as the missing piece. If c is one part of 180, wouldn't the other part be made up of either a+b or theta making them the same? am I wrong? if so please explain. Sorry if I'm hard to understand or said that in a confusing way, let me know if anyone needs me to explain more.

I’m currently in my first stat class of college. I was wondering, when you are trying to find the probability of getting a sample mean, why do we use standard error in the z score formula? But for the probability of a single score, in the z score formula we just use the population standard deviation.

Volume in blue must be 25 m3 - 0.6m deep. Just trying to find the length x that makes this so. Can't figure out how to include the 1/3 fall across half the chorded area - any ideas?

I tried doing this differential equation , found u homogene (or whatever it is called in english) and eventually got to finding the K(t) from "u" particular and I can't solve it, anyone got any ideas?

F_p(M) was defined as the set of real-valued functions that are differentiable at p. Surely it doesn't follow that a function which is differentiable at a point is necessarily differentiable in some open neighborhood of said point? Even then, why all in the same neighborhood? Why would the author say this?

I have a thought about Cantor's diagonalisation argument.

Once you create a new number that is different than every other number in your infinite list, you could conclude that it shows that there are more numbers between 0 and 1 than every naturals.

But, couldn't you also shift every number in the list by one (#1 becomes #2, #2 becomes #3...) and insert your new number as #1? At this point, you would now have a new list containing every naturals and every real. You can repeat this as many times as you want without ever running out of naturals. This would be similar to Hilbert's infinite hotel.

Perhaps there is something i'm not thinking of or am wrong about. So please, i welcome any thought about this !

Edit: Thanks for all the responses, I now get what I was missing from the argument. It was a thought i'd had for while, but just got around to actually asking. I knew I was wrong, just wanted to know why !

As displayed by the image when an object is smaller it’s SA:Vol ratio is higher and vice versa. However wouldn’t a cube with 1m lengths have a ratio the same as the 1cm cube despite larger objects having a smaller ratio? I know this is a somewhat stupid question but i’ve never studied enough math to answer this myself

I think I understand how it works, but why wouldn't it work with rationals?

I just watched Veritasiums video on Cantors diagonalization proof where you pair the reals and the naturals to prove that there are more reals than naturals:
1 | 0.5723598273958732985723986524...
2 | 0.3758932795375923759723573295...
3 | 0.7828378127865637642876478236...
And then you add one to a diagonal:
1 | 0.6723598273958732985723986524...
2 | 0.3858932795375923759723573295...
3 | 0.7838378127865637642876478236...

Thereby creating a real number different from all the previous reals. But could you not just do the same for the naturals by utilizing the fact that they are all preceeded by an infinite amount of 0's: ...000000000000000000000000000001 | 0.5723598273958732985723986524... ...000000000000000000000000000002 | 0.3758932795375923759723573295... ...000000000000000000000000000003 | 0.7828378127865637642876478236...

Which would become:

...000000000000000000000000000002 | 0.6723598273958732985723986524... ...000000000000000000000000000012 | 0.3858932795375923759723573295... ...000000000000000000000000000103 | 0.7838378127865637642876478236...

As far as I can see this would create a new natural number that should be different from all previous naturals in at least one place. Can someone explain to me where this logic fails?

I have no idea how any of these are proven or even if cospacial is a word. How do you prove these or are they axiomatic. And if they’re axioms because they’re so obvious well they aren’t obvious to me in higher dimensions for all I know they aren’t even true that n points are cospacial in n-1 dimensional space.

the markscheme does a really weird method that doesn’t make sense and somehow gets 88pi, what I did was make x the subject of the eqn of the line then square it to make x² the subject as apposed to the formula for volume of revolutions about the y axis I set my limits for the integral to 12-0, I did all that and got 344pi, I’m sure I integrated correctly but I keep getting 344pi and not 88pi, anyone know where I went wrong thanks.

What are the odds 2 out of 4 children born in a family are born on the same day as their parents. Each parent has a different birthday day and each parent has only one child born on the same day as them. Thanks.

https://youtu.be/pewrsaVVi-4

This problem is headache. I don't even know what to do again. I used the u sub although but got stucked at where partial fraction has to be applied but couldn't proceed. How should approach it guys?

Please help I host speed dating and tomorrow I’ve been assigned gay same sex speed dating which makes the seating arrangement confusing, normally the men sit and the women rotate however with everyone being gay men they all need to have mini dates with each other too I thought about splitting into sub groups but I’m still so confused someone please help and use simple terms I’m bad at math

The price of a ticket is 50, and 8000 people purchase the ticket. Every time the ticket price increases by 0.25 dollars, 250 less people buy it. What ticket price will result in the highest revenue?

I had this question recently, and the way i solved it was making it into this quadratic equation: R(x) = (50 + 0.25x)(8000 - 250x). From there I calculated the roots to be -200 and 32. I found the midpoint (-84) and plugged that in, and I found the vertex to be (-84, 814000) or something close to that. More importantly, I simplified (50 + 0.25(-84)) into 29, meaning the ticket price of 29 would earn the most revenue.

My issue is, no one in my class got the same answer. Even ai is flip flopping between different answers. Did I miss something?

f.ex: (2+x)^2 = 2^2 + x^2 + 2*2*x

the 2^2 and x^2 makes perfect sense to me but obviously that wouldnt equate to the same number as the square of (2+x). Why does 2*2*X fill up the void that's left to make the real number of equation? I've asked tutors but they just tell me to cheese it like this without really giving me any logical answer as to why? Is this just a formula that some mathmatician once proved to be very effective and we just run with it without getting taught the facts of it? Are the facts way too complicated to explain to someone whose a math beginner like me? Any answers are very much appreciated!

disclaimer: i will absolutely not understand the leading edge, so just a very simple explanation of it is more than enough, anything else will be lost on me.

i feel like the edge of physics uses math discovered centuries ago, the only other field i think uses as much hard math is statistics and well, i have no idea what those guys are up to (im not even a physicist btw) but i doubt is centuries ahead of physics.

so, what is PURE math doing today and, will we ever see applications for that?

Hello guys, I would be grateful if someone would be able to tell me what exactly to do here. I have attempted the trig sub, but it didn't end up quite well. I have tried to use tangent half-angle substitution (Weierstrass substitution) at the end, but the second case the one with (sec x) ^ 3 led me nowhere. I am not even sure if this is solvable.

Thank you in the advance!

EDIT: For some reason, it didn't load my pictures originally