Product Definition and 1000 Threads

Homework Statement Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible. Homework EquationsThe Attempt at a Solution We know that A is invertible, so there exists a matrix C such that CA = I. Then we can right -multiply by B so that CAB...

Problem: Fix some vector ##\vec{a} \in R^n \setminus \vec{0}## and define ##f( \vec{x} ) = \vec{a} \cdot \vec{x}##. Give an expression for the maximum of ##f(\vec{x})## subject to ##||\vec{x}||_2 = 1##. My work: Seems like a lagrange multiplier problem. I have ##\mathcal{L}(\vec{x},\lambda)...

Homework Statement I am asked to offer an example of two commuting elements whose product does not have an order equal to the least common multiple of their individual orders. Homework EquationsThe Attempt at a Solution Consider ##-1## and ##1## in ##\mathbb{Z}##. Then ##1+(-1) = 0## which...

Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...

The first part of the problem was easy enough as was finding the directions of the other two, but I am having trouble finding the correct angle for the last two cross products in order to find the magnitude.

I have a question about calculating solubilities of sparingly soluble salts. Eg Ksp CaF2 = 4 x 10-11 So, Saturation Index of CaF2 is: SICaF2 = IP / Ksp Where IP = Ionic Product = [Ca2+] x [F-]2 [Ca2+] and [F-] are molar concentrations of each ion. Example: We have 400 ppm Ca and 12 ppm...

Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...

Homework Statement A\cdot B\times C\quad =\quad 2\\ (2A+B)\quad \cdot \quad [(A-C)\quad \times \quad (2B+C)]\quad =\quad ? Homework Equations Various cross product and dot product properties The Attempt at a Solution I've only managed to get so far, don't really know what to do next A\cdot...

Hi, For school we are currently working with heterogeneous equilibria. I am given a salt that will be solved in water and I have to calculate the concentrations of the ions. I have to use the solubility product for this. In the Netherlands we are provided with a reference book that has all...

I am trying to derive the law of signs from the cross product. First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go...

Homework Statement Verify the identity: ## \nabla \times ( A \times B) = (B\bullet \nabla)A - (A\bullet\nabla)B + A(\nabla \bullet B)-B(\nabla\bullet A)## My issue here is I don't understand the significance of why a term has B or A on the left of the dot product, and another has B or A on...

I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...

Homework Statement Hello, We know that if A and B are two unbounded densely defined operators, it does not mean that AB is also densely defined. But if A is bounded then D (AB) = D (B) ie AB is densely defined. Is AB densely defined if: 1) B is bounded and A is unbounded densely defined...

Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##. With the construct of a triangle, the Law of Cosines is proved. ##\mathbf A## points to the tail of ##\mathbf B##. Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...

I have seen a proof for the formula of A.B = ||A|| ||B|| cos(theta)[ proof using the diagram and cosine rule]. In the proof they have assumed that distributive property of dot product is right. diagram is given below c.c =(a-b).(a-b) = a^2 +b^2 -2(a.b) [ here they used distributive law] I...

Dear all, I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted. For instance, the Hilbert space ##W## of a larger system is spanned by the direct product of...

Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?

Why A .A =||A||^2(dot product) in vector analysis. Every where in vector analysis mathematicians used this result. To prove A.B = ||A|| ||B||cos(theta) we assume that A.A is ||A||^2 , without assuming this we can't prove A.B = ||A|| ||B||cos(theta) . I think they assumed it because dot product...

Homework Statement Consider the vector space of all continuous functions on the interval C[-1,1]. That is V = C[-1,1] show that <f(x),g(x)> = ∫(-1,1) x2f(x)g(x)dx defines an inner product on C[-1,1]. I have shown <f,g> = <g,f>, <kf,g> = k<f,g>, <f+g,h> = <f,h> + <g,h> and I am trying to...

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

I do not know if this is the proper rubric to ask this question, but I picked the one that seemed the most relevant. I have noticed some superficial resemblance between the tensor product and the ultraproduct definitions. Does this resemblance go any further? While I am on the subject of...

Prove $$\left(1+\frac{1}{\sin x}\right)\left(1+\frac{1}{\cos x}\right)\gt 5$$ for $$0\lt x \lt \frac{\pi}{2}$$.

Are there such companies that specialize in making existing products reliable and safe. Thanks. P.S. I have a patent that I need to improve.

svein submitted a new PF Insights post Reflections on Product Quality Continue reading the Original PF Insights Post.

The problem: By considering w x (p x w) resolve vector p into a component parallel to a given vector w and a component perpendicular to a given vector w. Hint: a x (b x c) = b(a x c) - c(a x b) I'm afraid I really have no idea where to go with this one. The hint leads to: p(w.w) - w(w.p) =...

Homework Statement Under which set of conditions is the ionic product of water, Kw, constant at a given temperature in aqueous systems? in dilute acidic but not dilute alkaline solutions.in dilute alkaline but not dilute acidic solutions.in both dilute acidic and alkaline solutions.only at...

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression: ## E = AB - B^*A^*## I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no...

Hi. Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically...

Homework Statement This is a child thread I'm creating from a previous topic: https://www.physicsforums.com/threads/combinatorics-problem.871661/#post-5473920 In that thread, I was helped to come up with the expression for the number of arrangements of R distinct types of objects given the...

We had integrals, so we have to have series as well. Here are 10 easy to difficult series and infinite products. Up to you to find out the exact sum. Rules: The answer must be a finite expression. The only expressions allowed are integers written in base 10, the elementary arithmetic...

Hello! I am reading something about applications of group theory in quantum mechanics and I got confused about the difference between direct sum and direct product. In many places I found that they mean the same thing. However, the ways I found them defined in the book I read from, seem to be...

Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.

Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...

Homework Statement This problem actually has 2 parts. For vectors a= (2,p,8) and b= (q,4,12), determine the values of p and q so that the vectors are a)perpendicular b) collinear textbook answer: a) p= 1 and q= -50 (answers may vary) b) p= 8/3 and q=3 Homework Equations a*b = 0 a*b =...

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

Homework Statement Using index-comma notation only, show: \begin{equation*} \underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} \end{equation*} Homework Equations...

Homework Statement Using index notation only (no expanding of terms), show: \begin{equation*} \text{(a) }\underline{ \bf{a}} \cdot \underline{\bf{A}} \underline{\bf{b}} = \underline{\bf{A}} \cdot \underline{\bf{a}} \otimes \underline{\bf{b}} \end{equation*} Homework Equations...

Homework Statement Using index notation only (i.e. don't expand any sums) show that: \begin{align*} &\text{(a) } \epsilon_{ijk} \det \underline{\bf{A}} = \epsilon_{mnp} A_{mi} A_{nj} A_{pk} \\ & \text{(b) } \det \underline{\bf{A}} = \epsilon_{mnp} A_{m1} A_{n2} A_{p3} \end{align*} Homework...

I was reading about Gauss's Lemma here: https://cims.nyu.edu/~kiryl/Algebra/Section_3.10--Polynomials_Over_The_Rational_Field.pdf Unfortunately, I am stuck on Lemma 3.10.1 that concludes that the product of a pair of primitive polynomials is itself primitive. I understand about how there is...

I am reading Donald S. Passmore's book "A Course in Ring Theory" ... I am currently focussed on Chapter 9 Tensor Products ... ... I need help in order to get a full understanding of the free abelian group involved in the construction of the tensor product ... ... The text by Passmore...

Homework Statement Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.) PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS Homework Equations T(v)=AT*v The Attempt at a Solution I was able to prove part a. I let v=(v1,v2)...

Hey! :o Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find for the ideal $I=2\mathbb{Z}$ of $\mathbb{Z}$ the $\{m\in M\mid am=0, \forall a\in I\}$ as a product of cyclic groups. We have the following...

Hello I found a bug in my code and can't figuring out the error. I tried debugging by showing the output of each variable step by step but I can't find my error. Here is what I have and what I want to do: I have a matrix A: 0000 0101 1010 1111 And I have a matrix B: 10000 21000 30100 41100...

(Not an assigned problem...) 1. Homework Statement pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product It follows that: 2. Attempt at a solution. I think that both of these solutions are...

1. Homework Statement For part a), I know that it is a nucleophilic addition reaction of CN- to form cynohydrin, but how come there are 2N atoms in E? How does NH4+ react with CH3CHO or the cynohydrin formed? For part b), what reaction is this? Homework EquationsThe Attempt at a Solution

I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...

Homework Statement A point P moves so that its distances from A(a, 0), A'(-a, 0), B(b, 0) B'(-b, 0) are related by the equation AP.PA'=BP.PB'. Show that the locus of P is a hyperbola and find the equations of its asymptotes. Homework EquationsThe Attempt at a Solution AP.PA' =...

1. Homework Statement Why is the product at anode Br2 for MgBr2? Instead of O2? Homework EquationsThe Attempt at a Solution [/B] The EΘ value for the oxidation of OH- is -0.40 V, And that of Br- is -1.07V And so, shouldn't oxidation of OH- be easier? And hence, O2 is formed at the anode?

Is there any proof for the matrixx formula of the cross product. I am asking this because I have seen many videos and they have used the matrixx formula and then proved that ||A X B|| = ||A|||B||sin(theta), khan academy also used the same method

Homework Statement Hi everyone, I am a first year physics student and we recently learned about torque. Every time I think I understand it something else comes up to confuse me - this time it is the direction. I tried looking in the forum and generally in google, but everyone only explains the...