I do not have a formal Theorem or Proposition I can reference. I have the feeling it's true, but obviously that's not proof-worthy! I browsed around the web at some theorems of systems of equations, and I did not find a Theorem or Proposition matching my claim.

Neat argument. Thanks for sharing! I will keep in mind not to forget the actual system of equations that the matrix represents.

I believe that my geometrical argument can be generalized from a line to a plane.

Consider the two equations

ax + by + cz = d
ax + ey + fz = g

for which it is given that they have the same solution set. Because the two equations have the same solution set, they define the same plane.

Properties of the first equation are

x-intercept: d/a
y-intercept: d/b
z-intercept: d/c

Properties of the second equation are

x-intercept: g/a
y-intercept: g/e
z-intercept: g/f

Because the equations define the same plane, their corresponding intercepts must be equal. Because the x-intercepts are equal, d/a = g/a, meaning that d = g. Because the y-intercepts are equal, d/b = g/e. By substitution d/b = d/e, meaning that b = e. Similar reasoning with the z-intercept shows that c = f.

Therefore they must be the same equation, and if you perform a Gaussian reduction you will get 0 = 0.

Is that logic correct? Could it not be extended to n-dimensions, as long as the equations share a leading coefficient such as the "ax" in this example?

Thank you for clarifying my mistake! I especially appreciate that you shared the intuition behind it, i.e. that I was assuming (2,3) to be a solution of the equation.

I suggest you look at the examples provided at the top of the Wiki lesson you linked in your first post. They go over the various scenarios where you may or may not get a 0=0 in a Gaussian reduction. The examples show that in general it's not so easy to draw conclusions from the 0=0 either showing up or not showing up. If you get a line like 0=7 then you definitely know the system is not consistent and therefore has no solution. But getting or not getting 0=0 means different things in different situations.


If you study the argument I gave you for what the (d-b)y = (e-c) is telling you here, I think you will understand why in this case the equation must be 0=0.


PairTheBoard

Your argument is that, if d-b were not equal to zero, then the system would have only one solution, contradicting the given that the second equation shares the entire solution set of the first. Then if the left side of the equation is zero, the right side must also be zero to give a true statement. Correct?

Does that reasoning extend beyond equations with just two unknows, such as equations with three unknowns like I mentioned in my previous post?

Thanks for discussing this with me!

The system having only one solution contradicts what we established up front. That being that the system has infinitely many solutions. Those solutions being the same as the solutions to the first equation. And also the same as the solutions to the second equation.

But why does it have to give a true statement? Why can't it be something like, 0=7 ? The reason is that if it were something like 0 = 7 that would mean the system of equations is inconsistent and therefore would have no solutions. But we know the system of equations has infinitely many solutions as discussed above. So that would be a contradiction. So the supposition, "if the rhs were non-zero" must be false.

Both arguments, are proofs by contradiction. You want to prove the proposition P. So you suppose ~P is the case and show it leads to a contradiction.




Why don't you set up a system of 3 equations in 3 unknowns with the leading coefficient the same for each. Eliminate the first variable in the second and third equations. Then the second and third equations become a subsystem with 2 equations in 2 unknowns. What do you do then? Why don't you try it and see what you get.


PairTheBoard

Yea, this is correct, although you are assuming b,c,e,f are non-zero. You can put your proof in algebraic terms by talking about solutions like (d/a,0,0) rather than as you say the "x-intercept".

However, I don't see how an easy argument could be made using Gaussian reduction on the system of 2 equations in 3 unknowns.

I believe what's true in general is: If two linear equations in n variables have the same solution set then they differ by at most a scalar multiple. As a corollary, if they share 1 non-zero coefficient then they are the same equation.

The proof would be along the same lines as the one you gave for n=3.


PairTheBoard

I could use some more Topology help.



I know the answer to the first one, but I just can't derive the formula. Professor enjoys saying, "You should know how to do this from Calc 3." I took it 15 years ago and did poorly then. I don't teach Pre-Calc, so my vector math is poor.

For the 2nd one, I'm in a similar boat, since stereographic projections are big on vector notation. Also, he didn't really like my proof by picture where I put a sphere in a cube and showed how to shoot from the North Pole to any point on the Unit Cube.

does anybody have experience with Simulink in matlab? should be basic but im having trouble generating my signal. I need a slope of 1, than a negative slope of one until its zero again. so like an upside-down v. trying to combine 3 ramp sources but I doesn't work for some reason.

I do. 3 ramp sources should be fine. Can you explain what's not working?

r(t) - 2*r(t-1) + r(t-2) should do the trick, no?

I just started using simulink, so I don't really know what I'm doing. However, my input setup was fine. When I tested it in a separate file with nothing but a scope it worked beautifully. However, when using it while trying to simulate a suspension it does not come out right, even though it's connected to see scope through a muxer only.
I know my suspension is probably wrong, it just seems weird that the input signal is messed up. Any ideas why that could be the Case?

Yo, guys, if I have to plot a graph by hand for a Curie point experiment for u = f(t), where u is voltage and t is temperature, are the values of u going to be on the ordinate (y axis) and of t on the abscissa (x axis)?

What does it mean to find two lists with specified initial values, given a dual array implementation of linked lists?

Idk specifically about the experiment you described but typically time is the x axis (independent variable).

Heat transfer question: we kind of rushed through radiation. We usually deal with "gray and opaque" surfaces. I understand opaque means the transmissivity is 0 but am unsure of "gray". I thought it just means that the frequency and angle do not matter, but at one point I wrote in my notes that gray means that the absorbisity equals the emissivity. That doesn't seem to make sense. Can someone clarify?

Look here

http://en.wikipedia.org/wiki/Emissivity and here http://en.wikipedia.org/wiki/Black_body

"Real materials emit energy at a fraction—called the emissivity—of black-body energy levels. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. A source with lower emissivity independent of frequency often is referred to as a gray body.[5][6] Construction of black bodies with emissivity as close to one as possible remains a topic of current interest."

You may read also the beginning of 2.1 page 9 on this book
http://books.google.com/books?id=7qC...page&q&f=false

(you may start reading the part of the first page here, a preview of the radiation of bodies, and then continue on the links )

You may benefit also from http://en.wikipedia.org/wiki/Diffuse_reflection

You really need to gradually understand all these concepts in order to appreciate what a variety of objects do actually.

Does this have anything to do with that all circles are squares in the Taxicab metric? Or with that the Taxicab metric is not the real-world, as-the-crow-flies metric? Or that it has multiple shortest paths between two points?

(BTW, I'm used to calling it the Manhattan metric, which I'm not sure is not better terminology.)

I'm not sure I understand what you're asking. Generally speaking, they are different mathematical tools with different applications. What its relationship is to the "real world" and why Euclidean geometry with its right angles and Pythagorean theorem has proved such a powerful tool worthy of generalization I'm afraid are questions which take me out of my depth.


PairTheBoard

For example, I would guess that why shortest paths are unique in one metric but not that other can be explained by the algebraic differences in the norms that induce them.

I was wondering how and what geometric differences can be explained by the fact that one norm is derived from an inner product.

Ah. Yea, I think that's an excellent question!

I know very little in way of an answer. For centuries Euclidean geometry was thought to be the only one possible and somehow dictated to us by the "real world". But in the 1800's they realized generalized geometries could be studied. e.g. the geometry on the surface of a globe. I know practically nothing about such geometries.



PairTheBoard

You may want to see also

http://en.wikipedia.org/wiki/First_fundamental_form

http://en.wikipedia.org/wiki/Second_fundamental_form

(and follow the sphere examples for instance)

Also

http://en.wikipedia.org/wiki/Metric_space

http://en.wikipedia.org/wiki/Metric_tensor

And continuing good reads for eg the weekend lol;

http://en.wikipedia.org/wiki/Elliptic_geometry
(http://en.wikipedia.org/wiki/Riemannian_geometry)

http://en.wikipedia.org/wiki/Hyperbolic_geometry


And a couple classic oldies books ( i may add more if i find online later but search any online differential geometry classes or youtube lectures on the respective terms. Also always a good idea to try on google books search for the references in all wikipedia links and i wont have to tell you of course that almost all these major books are illegally available on usenet too if you cant buy them or get from libraries)

http://books.google.com/books?id=hkE...page&q&f=false

https://books.google.com/books?id=db...page&q&f=false

or a more recent easier one;

https://books.google.com/books?id=ag...essley&f=false

Can you offer an answer to this question?

Does the decimal representation of 3^(1/2) =1.732050808.......have a place n+1 digit that the first n digits are repeated (and then it changes of course etc as its irrational) (starting at n+1 digit say ie if n =5 the first are 17320 and they are not repeated clearly.

How about the first log(n) digits?

Example the first 200 digits are;

1.732050807568877293527446341505872366942805253810 38062805580697945193
30169088000370811461867572485756756261414154067030 29969945094998952478
81165551209437364852809323190230558206797482010108 46749232650

nowhere of course the first log 200~5 "17320" digits are repeated...(same for any other n lower than 200)

In a restaurant, if there are 6 different options on 4 courses..so 24 options. Each person needs to order 4 courses, 1 from each course, no exceptions. There are 9 people at the table. What are the chances of them ordering the same thing and how did you get to your conclusion? Thanks in advance!

of all nine of them ordering the exact same four-course meal?
the chance the second diner order the same thing as the first is (1/6)(1/6)(1/6)(1/6). then keep multiplying this for the next seven people.

so ((1/6)^4)^8 = some very small number.

Same order yes. So 6 options, 4 times. 9 people all choose the same.

It strongly depends on what the options are. Also whether the 9 diners are all hipsters or Scottish.

Lol...that's funny...let's remove the popularity of an item and just say 6 equally popular items on 4 courses.

Seems it should be ((1/6)^4)^9 no?