# Search results for query: *

1. ### Determinant problem

Thank you for your help !
2. ### Determinant problem

I think I get it, but it is a little difficult: ## \begin{align*} \det B &= \sum_{\sigma\in S_n} \epsilon(\sigma) b_{\sigma(1)1}\cdots b_{\sigma(n)n}\\ &= \sum_{\sigma\in S_n} \epsilon(\sigma) a_{\sigma(1)1}\cdots a_{\sigma(n)n} (-1)^{\#S_\sigma} \end{align*}## where set ##S_\sigma =...
3. ### Determinant problem

I have received an explanation in terms of permutations, but I didn't get it. That's why I tried to find another way. What do you mean ?
4. ### Determinant problem

The problem with the determinants in the cofactors is that neighbor columns have the same sign and break the 'chessboard' structure. That's why I reverse the sign of the ##k-1## first columns, in order to recover this structure, which allows me to apply ##{\cal P}(n-1)##.
5. ### Determinant problem

Homework Statement Given a matrix ##A = (a_{ij})##, we define matrix ##B = \begin{pmatrix} a_{11} & - a_{12} & a_{13} & \cdots \\ - a_{21} & a_{22} & -a_{23} & \cdots \\ a_{31} & - a_{32} & a_{33} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{pmatrix}##. Another way to define ##B## is...
6. ### 3D geometry parallelepiped problem

Hi, here is the picture
7. ### 3D geometry parallelepiped problem

Homework Statement [/B] Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity. I think I have understood the problem, could you verify my demo please ? Thanks Homework Equations The Attempt at...

Ok, thanks !
9. ### Displacements on a grid

Homework Statement We fit the plane with a coordinate system, and we consider the set of points with coordinates in ##\mathbb{N}\times\mathbb{N} ##. To link two points in this coordinate system, we only allow unit displacements, and only increasing displacements. In how many ways can one...
10. ### Linear Transformation and Inner Product Problem

Hint: the expression 'standard representation of ##T##' means 'representation of ##T## in an orthonormal basis'.
11. ### Complex numbers on the unit circle

The problem statement states that all three points belong to the unit circle, so ##z_2## and ##z_3## can be deduced from ##z_1## by a rotation centered at the origin. That is to say there exists angles ##\theta## and ##\rho## to determine such that ##z_2 = e^{i\theta} z_1 ## and ## z_3 =...
12. ### Complex numbers on the unit circle

Imagine that you had to prove the reciprocal, that is, if ##ABC## is equilateral, then ##z_1 + z_2 + z_3 = 0##. This can be solved by an elementary geometric argument : ##\angle AOB = \angle BOC = \angle COA = 2\pi / 3##. So ## z_2 = e^{2i\pi/3} z_1 ## and ## z_3 = e^{2i\pi/3} z_2 ##. The cubic...
13. ### Complex numbers on the unit circle

Consider points ##A(z_1)##, ##B(z_2)##, and ##C(z_3)##. Triangle ##ABC## is equilateral iff ##C## is deduced from ##B## by a rotation of center ##A## and angle ##\pi / 3##. How do you write such transformation ?
14. ### Prove the function is Riemann-integrable

But why would one want to prove integrability of a step function defined on a segment ? Such functions are integrable by definition. It is like if someone said 'Let ABCD a rectangle. Show that angle ABC equals 90 degrees'. I don't get the subtlety.
15. ### Prove the function is Riemann-integrable

I don't understand the problem statement. The integral of a step function defined on a segment is a fundamental definition rather than a property. It seems to me that you are asked to prove a definition, which seems to be nonsense to me.
16. ### Geometry question -- several lines through parallelograms

@theBin: Honestly you are attracting attention for nothing and I don't like it. As I said, it is a high school problem, and anyone that knows two or three things about line intersections can get to the same result I got to, in minutes, following the process described in a previous post. So...
17. ### Geometry question -- several lines through parallelograms

@theBin : it is just a high school problem that I tried to brush up on my geometry, so it doesn't need advanced geometry to be solved. However, you say that the 'or parallel' statement is wrong, but it's not what I find (by the means I used).
18. ### Geometry question -- several lines through parallelograms

Intuititively, I understand the idea, but I don't immediately see how it simplifies the problem. In my mind, it amounts to replace the word 'parallelogram' by the word 'rectangle' in the problem statement.
19. ### Geometry question -- several lines through parallelograms

What do you mean ? You may give your detailed solution as the exercise is already solved, it's just that I really feel that mine is longer than it needs to be. It consists in fitting a coordinate system on the parallelogram, finding the cartesian equations of (EB), (HD), and (IC), and start...
20. ### Geometry question -- several lines through parallelograms

Hello. Thank you for your post. What does parallelogram AHGD less parallelogram ABFE mean ? Can you draw a picture ? What should be understood by orientation of several parallelograms ? I don't understand this. It is possible after all that I understood nothing of what you said :-), but...
21. ### Geometry question -- several lines through parallelograms

Homework Statement On the picture, ##ABCD## is a parallelogram, ##(EF) // (AB) ##, and ##(GH) // (BC)##. The problem is : show that lines ##(EB)##, ##(HD)##, and ##(IC)## either all meet in ##M##, or are parallel. Homework Equations The Attempt at a Solution I've solved the problem...
22. ### Minimum of piecewise defined function

I don't know, post it in your language and hopefully someone who understands it will tell you. Good luck.
23. ### Minimum of piecewise defined function

I wouldn't say incorrect, because the ideas are there, but the wording is so strange that it looks like you don't understand. Firstly because you start a formal proof by writing "By intuition", which I find very hard to swallow. Secondly because you conjecture something that obviously exists.
24. ### Minimum of piecewise defined function

And I have tried the best of my best to help you :wink:
25. ### Minimum of piecewise defined function

No, you don't conjecture that ##n\to 1/2n\pi## exists! It does exist since it is perfectly defined as soon as ##n > 0##! The only thing you conjecture is that their exists a sequence that will at a time or another fall within ##[0,b[## because it approaches 0 by the right side, and at a time or...
26. ### Minimum of piecewise defined function

The question was to show that in any interval in the form ##[0,b[##, there exists ##x## such that ##f'(x) < 0##. What is interesting with this question is that one can use the computer in order to get an intuition of the answer. When you graph the function and see what looks like endless...
27. ### Minimum of piecewise defined function

You should observe that ##f'## oscillates about the x-axis as you zoom in around ##0^+##, at multiple orders of magnitude, meaning that the sign of ##f'## alternates as you get closer to zero. One can think that these oscillations occur indefinitely, this is to say that the sign of ##f'##...
28. ### Minimum of piecewise defined function

Ok for the global minimum. For Q.b, I insist, graph the function with a visualization tool, and zoom in around ##0^+##, at different orders of magnitude. What do you see ? What can you conjecture ? How does sequence ##x_n## prove that conjecture ?
29. ### Minimum of piecewise defined function

Q.a : I agree with the proof of continuity you've given when ##x\neq 0##. You didn't prove differentiability when ##x\neq 0## but a similar argument will do. For continuity and differentiability in 0, I believe you are jumping to a conclusion. You assume (correctly) that ##x^4 \sin(\frac{1}{x})...
30. ### How to find upper bound for recurrence relation

I don't understand. If one consider that the indexation of sequence ##T(n)## is in ##\mathbb{N}-\{0\}##, then the index ##n## must be a multiple of ##6##, since 2 and 3 are mutually prime divisors of ##n##. At first sight you must consider a sequence of type ##T(6n) = T(3n) + T(2n) + c ##. But...
31. ### Using series to prove hypothesis of right triangle's < limits

You just need to understand the geometric interpretation of complex transformation ##z\to az+b## when ##a## and ##b## are complex numbers. When it is not a translation, it is a similarity (find its center, its angle, its ratio). Then you will understand the geometric interpretation of ## (z' -...
32. ### Using series to prove hypothesis of right triangle's < limits

The transformation of point ##A## to point ##P_{n+1}## is for every ##n\ge 1## determined by a similarity of center ##P_n##, angle ##-\pi / 2##, and ratio of similarity ##r_n##, which translates in complex coordinates by the sequence ## P_{n+1} - P_n = r_n e^{-i\frac{\pi}{2}} (A - P_n) ##...
33. ### Using series to prove hypothesis of right triangle's < limits

One way to do this is to work in the complex plane. You can start with the following sequence defined for ##n\ge 1## by : ## P_{n+1} - P_n = r_n e^{-i\frac{\pi}{2}} (A - P_n) ## ##P_1## given Taking the modulus, you find the following expression for ##r_n = \frac{2^{n-1}}{|A-P_n|} ## You can...
34. ### Non singular matrix M such that MAM^T=F

Let's say that ##A## is a real matrix. If ##A## is anti-symmetric (## A^T = -A##), then ##A^2## is symmetric. The spectral theorem says it is diagonalizable in an orthonormal basis. Could the diagonal matrix be a representation of ##F^2## in another orthonormal basis ? I don't know, it is just...
35. ### Non singular matrix M such that MAM^T=F

I feel that it is a way or another related to the spectral theorem. Try to work on ##A^2##.
36. ### Prove/disprove these subgroups

I understand ##S_n## is the symmetric group. What is ##D_n## ?
37. ### Isomorphism is an equivalence relation on groups

Yes, just for easier reading, not to criticize you
38. ### Isomorphism is an equivalence relation on groups

When you describe a group, you have to specify a pair ##(S,\star)##, where ##S## is a set, and ##\star : S\times S \to S ## is the composition law ( or internal law ).
39. ### Isomorphism is an equivalence relation on groups

Yes if ##f## is a group isomorphism, so is its inverse. However, you don't explicitly assign internal laws in your notations, and it can be bothering for the reader.
40. ### Arbitrary Union of Sets Question

Ok so you infered that the union was equal to ##\mathbb{N}##. How do you prove that two sets are equal ?
41. ### Arbitrary Union of Sets Question

The problem statement is incomplete, it doesn't describe ##A_n##. In terms of notation, ##x \in \cup_{n\in\mathbb{N}} A_n ## if and only if there exists ##n\in\mathbb{N}## such that ##x\in A_n## (##x## belongs to at least one of the ##A_n##). ##x \in \cap_{n\in\mathbb{N}} A_n ## if and only if...
42. ### Double integral solution

The integrand ##f(x,z)## is undefined when ##x=4##. You will have to discuss integrability of ## x \to f(x,z) ## on ##[0,4[##, and of ## z \to \int_0^4 f(x,z) \ dx ## on ## [0,5]##. If you can justify this, your double integral is well-defined and you can evaluate the integrals in any order you...
43. ### Plotting the linear system

I didn't check the correctness of your result, but assuming all is right : If it has a unique solution, so it's a point. That means that the solution verifies ## x+ y + z - 1 = 0 ## when ##x,y,z## describe ##\mathbb{R}^3##. It's a plane passing through (1,0,0), (0,1,0), and (0,0,1). We...
44. ### Parametric Equations of Tangent Line

You want to find the tangent line as the intersection of plane ## P_1 |\ x = 2 ## and plane ## P_2 |\ a x + by + cz + d = 0 ##, where ## (a,b,c) = \text{grad }F(2,1,15) ##, ##F(x,y,z) = 2x^2 + 5y^2 -z + 2 ##, and ##d## determined by the fact that ##(2,1,15)## belongs to ##P_2##. In cartesian...
45. ### Linear system of equations

I don't want go off topic so I'll just explain why your solution set is a line. A line passing through point ##A## and directed by a vector ##\vec v## is the set of points ##M## such that vector ##\vec {AM}## and ##\vec v## are colinear. Your solution set can be written ##S = \{ A + z \vec v, \...
46. ### Linear system of equations

Your solution set ##S## (I see it was confirmed correct by Mark44) is a parametrization of line, i.e it can be written ## A + z \vec v ## where ##A = (8,6,0,1)## and ##\vec v = (-7,-5,1,0)##
47. ### Linear system of equations

Assuming your solution is correct, ##S## is a line of ##\mathbb{R}^4## passing through ##(8,6,0,1)## and directed by vector ##(-7,-5,1,0)##. It is not a vector space but an affine subspace ##\mathbb{R}^4##
48. ### Sum involving reciprocal of binomial coeffients

And if you write ##f(r^\frac{1}{r}) = \sum_{r=0}^n \sum_{j=1}^r \frac{1}{C(n,r) }## and exchange the order of summation ? You will have ##f(r^\frac{1}{r})## function of ##a_1,...,a_n## ?
49. ### Area bounded by two functions

It bugs me that you have explained things so many times but still, it's not crystal clear for me what you are doing. I can't track your ideas in details and it bugs me. Maybe the problem is in communication of ideas. Have you ever observed the precision of the staff mentors on this forum? In a...
50. ### P-series converges?

Mr Vickson answers your question "why the harmonic serie diverges?" by comparing the harmonic serie to an integral. The conclusion of it is that the partial sum ##H_N = \sum_{n=1}^N \frac{1}{n}## is equivalent as ##N \to +\infty## to ##\ln N##. This means that as ##N\to \infty##, ##H_N / \ln N...