Proof Definition and 999 Threads

Hello all, I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the row space to the null space column space to the left null space ------------------------------------------------ In the book of G. Strang there is this nice picture...

In Phillip Harris' (U. Sussex) post on special relativity he includes on p. 45 an algebraic proof of invariance of spacetime intervals. He starts with the definition S^2 =c^t^2 - x^2 -y^2 -z^2, he inserts the Lorentz transform expressions fot t and x, and he does some algebra to show that one...

Homework Statement:: Show that for every real number ##x## there is exactly one integer ##N## such that ##N \leq x < N+1##. (This integer is called the integer part of ##x##, and is sometimes denoted ##N = \lfloor x\rfloor##.) Relevant Equations:: N/A I have tried reading the solution given...

I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer. The way I thought of the problem: I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...

I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...

N.B. I have inserted the proof here as reference. See the bolded text. My question is, isn't the reasoning "##x^{2}+5 \varepsilon<2,## thus ##(x+\varepsilon)^{2}<2 .## " circular? If we can already find an ##0<\varepsilon<1## such that ##x^{2}+5 \varepsilon<2,## Can't one also claim that " we...

In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...

Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...

If P = NP, then the solution of any decision problem in NP can be found efficiently. Consider the Pythagoras theorem. It can be represented as an encoded (binary) string. (How?) Let D be the decision problem asking whether the input to D is a proof of the Pythagoras theorem. Given an...

I searched through the courses but I can't find any formula to help me prove that the expression is an eigenfunction. Am I missing something? What are the formulas needed for this problem statement?

hi guys in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass : $$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$ is there is any proof for this definition ? and what is the approach for it

I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?

In the following proof: I didn't understand the following part: Isn't it supposed to be : ## a > s_A - \epsilon >0 ## and ## b > s_B - \epsilon >0 ## Because to prove that ## s ## is a supremum, we need to prove the following: For every ## \epsilon > 0 ## there exists ## m \in M ## such...

Reorder the statements below to give a proof for G/G\cong \{e\}, where \{e\} is the trivial group. The 3 sentences are: For the subgroup G of G, G is the unique left coset of G in G. Therefore we have G/G=\{G\} and, since G\lhd G, the quotient group has order |G/G|=1. Let \phi:G/G\to \{e\} be...

Summary:: x Hey, I'm learning calculus and had to prove the following Lemma which is used to prove AM-GM inequality, I had tried to prove it on my own and it is quite different from what is written in my lecture notes. I have a feeling that my proof of the Lemma is incorrect, but I just don't...

The potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m}...

hi guys i am trying to follow a proof of the generalized uncertainty principle and i am stuck at the last step : i am not sure why he put these relations in (4.20) : $$(\Delta\;C)^{2} = \bra{\psi}A^{2}\ket{\psi}$$ $$(\Delta\;D)^{2} = \bra{\psi}B^{2}\ket{\psi}$$ i tried to prove these using the...

A cylinder contains an initial volume V1 = 1m^^3 of a perfect gas at initial pressure p1 = 1 bar, confined by a piston that is held in place by a spring. The gas is heated until its volume is doubled and the final pressure is 5 bar. Assuming that the mass of the piston is negligible and that the...

I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...

"Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product." I'm taking...

##A ∖ B## can't include any elements that are not in ##A##, so it is the same as saying ##A∖(A∩B)##; it's exactly the elements of ##A## except those in ##A∩B##. ##A∖(A∖(A∩B))## is exactly the elements of ##A## except those in (exactly the elements of ##A## except those in ##A∩B##). This is the...

##\sqrt[n]n!## ≤##\frac{n+1}{2}## ##\frac {1}{n}## log [n!]= log##\frac{n+1}{2}##

Theorem “Identities of + are unique”: O₁ = O₂ Proof: O₁ = Left Identity of + O₁ + x I'm a little confused where to begin this proof, I don't know if that is the first step either I think it is. Proofs are not a strength of mine so I struggle to see how to show that O₁ = O₂. Any guidance would...

First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...

I can not follow this mathematically. I am guessing one of the signs is incorrect? Can anyone verify ?

this is a solution posted by my colleague, i have a problem in understanding how he got to conclude on equation (ii) is this not supposed to be ##(k+2)!≥ 2^{k+1} (k+1)## ##(k+2)(k+1)!≥ 2^{k+1} (k+1)## ?...

Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...

Here is the proof I was reading: https://mathschallenge.net/full/irrationality_of_pi I have a question about this very last inequality at the end: How did they get that "less than 1" bit? .

Summary:: I'm reading Adkins' book "Algebra. An approach via Module Theory" and I'm trying to prove theorem 3.15 In theorem 3.15 of Adkins' book says: Let ##N \triangleleft G##. The 1-1 correspondence ##H \mapsto H/N## has the property $$H_1 \subseteq H_2 \Longleftrightarrow H_1/N \subseteq...

I was watching a lecture that made the conclusion about the torsion being equal to zero necessitated that the path was planar. The argument went as follows: -Torsion = 0 => B=v, which is a constant -(α⋅v)'=(T⋅v)'= 0 => α⋅v= a, which is a constant (where α is a function describing the path and...

Note that if we prove problem 4, the proof for problem 5 follows directly. We use properties of logarithms to combine the right hand side of ln into a single logarithm. Then we raise both side of the inequality to a power of e. Which leads us to the desired inequality. But, when I try to be...

Find a graph to a number $\delta$ such that $$\textit{if } |x-1|<\delta \textit{ then } \left|\dfrac{2x}{x^2+4}-0.4\right|<0.1 $$ ok I always had a very hard time doing these I did look at some examples but still ? did a ibispaint drawing to start basically it looks like we are finding the...

Show by combinatoric means that ##p\,|\,(a^p-a) ## for a prime ##p \in \mathbb{P}## and a positive integer ##a\in\mathbb{N}##. Hint: e.g. consider chains of ##p## coloured pearls where ##a## is the number of colours.

Well, it is what I consider the most basic proof I've ever seen. I like it for several reasons: it is so simple. that you can convince a preschool kid, it uses one of the most fundamental principles at all: you cannot derive false from true, and it is old enough to count as such: No, sorry...

Math Proof Training Camp and Practice Guidelines We frequently get questions on how and where to learn to prove statements. The vast majority of scientists learned them by doing. It is a try and error approach trained in tutorials and exams. This forum is meant to practice these techniques on a...

I'm studying the proof of this theorem (Zorich, Mathematical Analysis II, 1st ed., pag.136): which as the main idea uses the fact that a diffeomorphism between two open sets can always be locally decomposed in a composition of elementary ones. As a remark, an elementary diffeomorphism...

From Zorich, Mathematical Analysis II, sec. 11.5.2: where as one can read from the statement, the sets could also be unbounded. I do not report here the proof of the fact a), beacuse I have no doubt about it and one can, without the presence of dark steps in the reasoning, assume a) as...

In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is...

As of 2020 is there any proof of tiny extra dimensions or observations of them?

find the general rule and prove by induction 1 = 1 1 - 4 = -(1 + 2) 1 - 4 + 9 = 1 + 2 + 3 1 - 4 + 9 -16 = -(1 + 2 + 3 + 4) I created this so far, but don't know if I am even going the correct direction

A square table of size 1001x1001 is filled with the numbers 1; 2; 3; ... ; 1001 in such a way that in every row and every column all those numbers appear. If the table is symmetric with respect to one of its diagonals, prove that in this diagonal all of the numbers 1; 2; 3; ... ; 1001 appear.

Hi.I have this trivial problem for a metric d(x,y) that d((x,y)≥0. My alternative proof is 2d(x,y)=√4d2(x,y)=√d2(x,y)+d2(y,x)+2d(x,y)d(y,x)=√(d(x,y)+d(y,x))2≥d(x,x)=0 .Well it perhaps is a trivial proof but I did not know of this proof so I wanted to post it. Do you know other alternative proofs...

Prove that $\cos\dfrac{\pi}{7}=\dfrac{1}{6}+\dfrac{\sqrt{7}}{6}\left(\cos\left(\dfrac{1}{3}\arccos\dfrac{1}{2\sqrt{7}}\right)+\sqrt{3}\left(\dfrac{1}{3}\cos\dfrac{1}{2\sqrt{7}}\right)\right)$.

Given: x\in A\cap B\leftrightarrow x\in A\wedge x\in B x\in A\cup B\leftrightarrow x\in A\vee x\in B x\in A-B\leftrightarrow x\in A\wedge x\notin B A=B\leftrightarrow(\forall x(x\in A\leftrightarrow x\in B)) Then prove using only the above and the laws of logic that: ™ (A\cup B)-(A\cap...

∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof. These are the relevant definitions listed in the book: Definition of Matrix of a Linear Map, M(T): Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1...

My question is how to show ##\mathcal{F} \subset \sum'##. Here is my work for the problem: Proof of hint: First we'll show ##\sum## is a monotone class. Let ##(A_n)_{n\in\mathbb{N}} \subset \sum## and ##F \in \mathcal{F}##. There are two things to verify. Suppose ##(A_n) \uparrow A =...

Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...

I read in one book proving one nature of variation(variation of high-order derivative). It writes that "##\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}##". But I don't understand where this ##F_0## comes out from.