Basis Definition and 1000 Threads

So while reading T. Frankel's "The Geometry of Physics", I was going through the part on cotangent bundles which ended with the definition of Poincare 1-form. The author argued that cotangent bundles are better suited than tangent bundles for some problems in physics and that there is no natural...

Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...

Need some sources on why PWM is widely used in inverters for DC - AC conversion applications, and their mathematical basis? Basically, I was wondering why inverters had to use PWM, instead of just getting a square wave of let's say a 50Hz frequency and just filtering out the odd order harmonics...

Hey! 😊 We consider the inner product $$\langle f,g\rangle:=\int_{-1}^1(1-x^2)f(x)g(x)\, dx$$ Calculate an orthonormal basis for the poynomials of degree maximum $2$. I have applied the Gram-Schmidt algorithm as follows: \begin{align*}\tilde{q}_1:=&1 \\...

Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this^^ (see the attached image). Here we sum over all modes m and 𝜙0 is a parameter. What will be a good set of basis for the...

If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4## Tried to use the integral...

If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B? I have calculated another basis of B, and found I can use linear combinations...

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...

I was reading these notes and then on page 23 I saw something a bit weird. Back in this thread I learned that ##\{ \partial_i \}## form a basis of ##T_p M##, and that a tangent vector can be written ##X = X^i \partial_i##, and it's not too difficult to show that components transform like...

A coordinate system with the coordinates s and t in $$R^2$$ is defined by the coordinate transformations: $$ s = y/y_0$$ and $$t=y/y_0 - tan(x/x_0)$$ , where $$x_0$$ and $$y_0$$ are constants. a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system is well...

a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...

Im very new to the terminologies of isometric basis and musical isomorphism, will appreciate a lot if someone could explain this for me in a simple way for a guy with limited experience in this field. The concrete problem I want to figure out is this: Given: Let ##v_1 = (1,0,0) , v_2 = (1,1,0)...

To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right? $$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$ $$\hat{v}_z = \hat{v} \cdot \hat{z}$$ And do that for...

Consider two entangled spin half particles given by the generic form of Bell Equation in Z-axis: ##\psi = (a\uparrow \uparrow + b\downarrow \downarrow)## where ##a^2+b^2=1## In a (2D) planer rotated (by an angle ##\theta##) direction the new equation can be given by: ##|\psi \rangle =...

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...

Why in ##l_2## space basis ##|1 \rangle=[1 \; 0 \; 0 ...]^{\mathsf{T}}##, ##|2 \rangle=[0 \; 1 \; 0 ...]^{\mathsf{T}}##, ##|3 \rangle=[0 \; 0 \; 1 \; 0...]^{\mathsf{T}}##... is called topological basis?

I'm reading a section on the derivative of a vector in a manifold. Quoting (the notation ##A^{\alpha}_{\beta'}## means ##\partial x^{\alpha}/\partial x^{\beta'}## - instead of using primed and unprimed variables, we use primed/unprimed indices to distinguish different bases): Now this "we know...

I think we can say that (u,v,u^v) is a positive basis, so as (w^v,v,w) and (u,w^u,w). (1) So u^v = βw v^w = γu w^u = λv where λ, β, and γ > 0 (*) (u^v, v^w,w^u) = (βw,γu,λv) \begin{vmatrix} 0 & 0 & β \\ γ & 0 & 0 \\ 0 & λ & 0 \\ \end{vmatrix} This determinant is positive by (*) What you...

On the way to Kruskal coordinates, Carroll introduces coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$$ {ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2 $$ ##R_s=2GM## and we're using a ##-+++## signature...

See this exercise: It ask for the matrix changing the basis E -> F If you pay attention, it write F in terms of E and write the matrix. Now see this another exercise: It ask the matrix B -> C, writing B in terms of C Which is correct? If it are essentially equal, where am i interpreting wrong?

Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...

Consider the transformation of the components of a vector ##\vec{v}## from an orthonormal coordinate system with a basis ##\{\vec{e}_1, \vec{e}_2, \vec{e}_3 \}## to another with a basis ##\{\vec{e}'_1, \vec{e}'_2, \vec{e}'_3 \}## The transformation equation for the components of ##\vec{v}##...

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...

We always can define a metric with a basis field ##g_{\mu\nu}=e_\mu \cdot e_\nu##, For a basis field ##e_\mu##, it can belong to a coordinate basis, then there is a corresponding coordinate system##\{x^\mu\}##,then we can have ##e_\mu=\frac{\partial}{\partial x^\mu}##, and ##[e_\mu , e_\nu]=0##...

Hello all, I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors e_1, e_2, e_3 in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows: e_i \times e_j =...

Hey! 😊 Let $1\leq n\in \mathbb{N}$ and $\mathbb{R}^n$. A basis $B=(b_1, \ldots, b_n)$ of $V$ is an orthonormal basis, if $b_i\cdot b_j=\delta_{ij}$ for all $1\leq i,j,\leq n$. Let $E=(e_1, \ldots,e_n)$ be the standard basis and let $\phi \in O(V)$. ($O(V)$ is the set of all isometries...

If this question is in the wrong forum please let me know where to go. For p, the vector space of polynomials to the form ax'2+bx+c. p(x), q(x)=p(-1) 1(-1)+p(0), q(0)+p(1) q(1), Assume that this is an inner product. Let W be the subspace spanned by . a) Describe the elements of b) Give a basis...

Hi, take a look at the picture from my textbook, specifically the bottom part: there are five lattice points, shouldn't that mean that there are also 5 "small basis balls"? Or can they be "shared"? If so, they are not all oriented in the same way - is that not important since there's no...

Hey! :o Let $1\leq k,m,n\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $U$ a subspace of $V$ with $\dim_{\mathbb{R}}U=m$. Let $u_1, \ldots , u_k\in U$ be linear independent. Show that there are vectors $u_{k+1}, \ldots , u_m\in U$ such that $(u_1, \ldots , u_m)$ is a basis of $U$. Hint: Use the...

For two different coherent states \langle \alpha|\beta \rangle=e^{-\frac{|\alpha|^2+|\beta|^2}{2}}e^{\alpha^* \beta} In wikipedia is stated https://en.wikipedia.org/wiki/Coherent_state"Thus, if the oscillator is in the quantum state | α ⟩ {\displaystyle |\alpha \rangle } |\alpha \rangle it is...

If propositions ##p,q\in{\mathscr L}_{\mathcal H}## (i.e., the lattice of subspaces of ##\mathcal H##) are incompatible, then ##\hat p\hat q\neq\hat q\hat p##. But since it's a lattice, there exists a unique glb ##p\wedge q=q\wedge p##. How are they mathematically related? In particular, I...

Fred H. Croom (Principles of Topology) and Tej Bahadur Singh (Elements of Topology) define local basis (apparently) slightly differently ... Croom's definition reads as follows:... and Singh's definition reads as follows: The two definitions appear different ... ... Croom requires that each...

On the basis of what brain is classified,like we say on the basis of function the neurons are of 3 types.

Hey everyone . So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking ! So i was...

As far as I know, a set of vectors forms a basis so long as a linear combination of them can span the entire space. In ##\mathbb{R}^{2}##, for instance, it's common to use an orthogonal basis of the ##\hat{x}## and ##\hat{y}## unit vectors. However, suppose I were to set up a basis (again in...

What's the meaning of the "bar" on the basis set of W at bottom right corner?

may someone explain to me or show me the steps of how equations 7a, 7b, 7c were determined from equations 5 and 6

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...

Hey! :o Let $1\leq n\in \mathbb{N}$ and $v_1, \ldots , v_k\in \mathbb{R}^n$. Show that there exist $w_1, \ldots , w_m\in \{v_1, \ldots , v_k\}$ such that $(w_1, \ldots , w_m)$ is a basis of $\text{Lin}(v_1, \ldots , v_k)$. I have done the following: A basis of $\text{Lin}(v_1, \ldots , v_k)$...

Hi PF! I'm trying to find the eigenvalues of this ODE $$y''(x) + \lambda y = 0 : u(0)=u(1)=0$$ by using the basis functions ##\phi_i = (1-x)x^i : i=1,2,3...n## and taking inner products to formulate the matrix equation $$A_{ij} = \int_0^1 \phi_i'' \phi_j \, dx\\ B_{ij} = \int_0^1...

I know these software packages were discussed a lot in the past, but I have not seen much input from the last couple years. I have used Matlab for many years, but remember using Maple in University Physics courses many years ago. I'm interested in a software package for symbolic math to use...

Hi All Given that the Riemann Curvature Tensor may be derived from the parallel Transport of a Vector around a closed loop, and if that vector is a covariant vector Having contravariant basis The calculation gives the result Now: Given that the Christoffel Symbols represent the...

Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...

Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!

You may be familiar with how you can express a vector field as a linear combination of basis vectors like so: X = Xi∂i Now, I know that normally, the basis vectors ∂i can be derived by taking the derivatives of the position vector for the coordinate system with respect to all the axes like...

If ##v## is an element of a vector space ##V## and for example ##\mathcal{B}=\{e_1,e_2,e_3\}## is a basis of ##V##, then, at least, there should be another basis for ##V## in which the vectors of ##\mathcal{B}## can be expressed, but at the same time, the vectors of this other basis must also be...

The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...

Let ## \begin{align}M =\begin{pmatrix} 2& -3& 0 \\ 3& -4& 0 \\ -2& 2& 1 \end{pmatrix} \end{align}. ## Here is how I think the JCF is found. STEP 1: Find the characteristic polynomial It's ## \chi(\lambda) = (\lambda + 1)^3 ## STEP 2: Make an AMGM table and write an integer partition...

It may be a valid argument that "A basis is not a property of an object. It's a choice humans make in the math for convenience. It makes no sense to say an object has or doesn't have a position basis or any other basis.". So state vector basis could be the map. object is the territory. But...

For the matrix $A=\left[\begin{array}{rrrrr} 1&0&0&4&5\\ 0&1&0&3&2\\ 0&0&1&3&2\\ 0&0&0&0&0\end{array}\right]$ Find a basis for NS(A) and $\dim{NS(A)}$ $\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{array}\right]=...