Basis Definition and 1000 Threads

In this lecture, the lecturer claims that the ##C^{\infty}##-module of smooth vector fields defined on a smooth manifold can lack to admit a basis (not even infinite dimensional). Indeed the set of smooth vector fields can be given an (infinite dimensional) vector space structure over the field...

In "A Student's Guide to Vectors and Tensors" Daniel Fleisch presents basis vectors and dual basis vectors like this: Then he writes: "The second defining characteristic for dual basis vectors is that the dot product between each dual basis vector and the original basis vector with the same...

My understanding is that the Hodge dual of a pseudo-form is always a "true" pseudo-form, and vice versa. However, I'm a little confused about how this applies to basis-forms in general. I believe I understand how it works for the ##0##-form case: the basis ##0##-form is the scalar ##1##...

Hello, I have watched a really good Youtube video on linear algebra by Dr. Trefor Bazett and it made me think about a question... () Personal Review A basis in 2D space is formed by any two independent vectors that are not collinear geometrically. Any vector in the 2D space can then be...

Hello, I am review some key linear algebra concepts. Let's keep the discussing to 2D. Vectors in the 2D space can be simplistically visualized as arrows with a certain length and direction. Let's draw a single red arrow on the page representing vector ##X##, an entity that is independent of the...

Theorem: The columns of A which correspond to leading ones in the reduced row echelon form of A form a basis for Col(A). Moreover, dimCol(A)=rank(A).

I think I can prove W is a subspace of V. I want to ask about basis of W. Let $$V = a_1+a_2 \sin t+a_3 \cos t +a_4 \sin (2t)+a_5 \cos (2t)$$ $$W = p(t) = q"(t) + q(t)$$ $$=-a_2 \sin t-a_3 \cos t-4a_4 \sin (2t)-4a_5 \cos(2t)+a_1+a_2 \sin t+a_3 \cos t +a_4 \sin (2t)+a_5 \cos (2t)$$ $$=a_1-3a_4...

I refer to the video of this page, where there is a description of Galilean relativity that is meant to be an introduction to SR, making the comprehension of the latter easier as a smooth evolution from the former. All the series is in my opinion excellent, but I think that this aspect is...

I'm reading Semi-Riemannian Geometry by Stephen Newman and came across this theorem: For context, ##\mathcal{R}_s:Mult(V^s,V)\to\mathcal{T}^1_s## is the representation map, which acts like this: $$\mathcal{R}_s(\Psi)(\eta,v_1,\ldots,v_s)=\eta(\Psi(v_1,\ldots,v_s))$$ I don't understand the...

Until now in my studies - matrices were indexed like ##M_{ij}##, where ##i## represents row number and ##j## is the column number. But now I'm studying vectors, dual vectors, contra- and co-variance, change of basis matrices, tensors, etc. - and things are a bit trickier. Let's say I choose to...

The 8 gluon fields of SU(3) can be represented (generated) by the 8 Gel-Mann matrices: $$ \lambda_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \lambda_2 = \begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \lambda_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0...

I would like to show that a LLL-reduced basis satisfies the following property (Reference): My Idea: I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought: So based...

I have a question about operators in finite dimension Hilbert space. I will describe the context before asking the question. Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle . Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...

Given an orthonormal basis ##\{e_1,\ldots, e_n\}## in a complex inner product space ##V## of dimension ##n##, show that if ##v_1,\ldots, v_n\in V## such that ##\sum_{j = 1}^n \|v_j\|^2 < 1##, then ##\{v_1 + e_1,\ldots, v_n + e_n\}## is a basis for ##V##.

If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|## ##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|## And we've a general vector ##|\alpha\rangle## such that...

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...

##f : [0,2] \to R##. ##f## is continuous and is defined as follows: $$ f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$ $$ f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$ ##V = \text{space of all such f}## What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...

Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved ## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...

Google top result says Toyota is researching Fluorine batteries that they claim will have 7x energy density of LiIon. However my textbook table of reduction potential gives lithium as higher than fluorine. Any idea what they base their claims on? Thanks Joe

I need to evaluate ##\nabla_{\mu} A^{\mu}## at coordinate basis. Indeed, i should prove that ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##. So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$ The first and third terms...

[Moderator's note: Spin off from another thread due to topic change.] I was thinking about the following: can we take as a basis vector a null (i.e. lightlike) vector to write down the metric ? Call ##v## such a vector and add to it 3 linear independent vectors. We get a basis for the tangent...

Can I conceive a countable position basis in Quantum Mechanics? How can I talk about the position basis in the separable Hilbert space?

Hi there, I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove : Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E## Then...

In textbooks, Bekenstein-Hawking entropy of a black hole is given as the area of the horizon divided by 4 times the Planck length squared. But the corresponding basis of the logarithm and exponantial is not written out explicitly. Rather, one oftenly can see drawings where such elementary area...

First of all, it is clear that we can find such a bases (the theorem is given in almost all of the books, but if you want to share some insight I shall be highly grateful.) We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as...

Divergence formula $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$ If we express it in terms of the components of ##\vec{A}## in unit basis using $$A^{*j} = \sqrt{g^{jj}} A^{j}$$ , we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}}...

What is the basis of 2x2 matrices with real entries? I know that the basis of 2x2 matrices with complex entries are 3 Pauli matrices and unit matrix: \begin{bmatrix} 0 & 1 \\[0.3em] 1 & 0 \\[0.3em] \end{bmatrix}, \begin{bmatrix} 0 & -i \\[0.3em] i & 0...

There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...

If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##. Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as $$ x =...

The term approximating basis is used by author Harry Floyd David is his book Fourier Series and Orthogonal Functions on page 56: So I have looked in other books on functional analysis, harmonic analysis...and even on Google and I cannot find any other text reference that uses this term. This...

Let ##S## be a set of all polynomials of degree equal to or less than ##n## (n is fixed) and ##p(0)=p(1)##. Then, a sample element of ##S## would look like: $$ p(t) = c_0 + c_1t +c_2t^2 + \cdots + c_nt^n $$ Now, to satisfy ##p(0)=p(1)## we must have $$ \sum_{i=1}^{n} c_i =0 $$ What could...

##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...

I put it in echelon form but don't know where to go from there.

I've a transformation ##T## represented by an orthogonal matrix ##A## , so ##A^TA=I##. This transformation leaves norm unchanged. I do a basis change using a matrix ##B## which isn't orthogonal , then the form of the transformation changes to ##B^{-1}AB## in the new basis( A similarity...

Hello, I have a question about the measurement of a qubit in the computational basis. I would like to first state what I know so far and then ask my actual question at the end.What I know: Let's say we have a qubit in the general state of ##|\psi\rangle = \alpha|0\rangle + \beta|1\rangle##. Now...

I have a given point (vector) P in R^3 and a 2-dimensional linear subspace S (a plane) which consists of all elements of R^3 orthogonal to P. The point P itself is element of S. So I can write P' ( x - P ) = 0 to characterize all such points x in R^3 orthogonal to P. P' means the transpose...

Hello! (Wave) We consider the usual representation of non-negative integers, where the digits correspond to consecutive powers of the basis in a decreasing order. Show that at such a representation, for the conversion of a number with basis $p$ to a system with basis $q$, where $p=q^n$ and $n$...

I am trying to do exercise 8.5 from Misner Thorne and Wheeler and am a bit stuck on part (d). There seem to be some typos and I would rewrite the first part of question (d) as follows Verify that the noncoordinate basis ##{e}_{\hat{r}}\equiv{e}_r=\frac{\partial\mathcal{P}}{\partial r},\...

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...

Hi everyone! I've been studying quantum mechanics for a while but I have a big big problem. If a system is in an eigenstate of energy (I use the eigenstate as a basis) it remains in this state forever. But if I describe the system with a different set of basis states (not eigenstates) the...

Are cartesian coordinates the only coordinates with a holonomic basis that's orthonormal everywhere?

It's probably more kind of math question. I consider a wave function of a harmonic oscillator, i.e. a particle in a parabolic well of potential. We know that the Hamiltonian is a Hermitian operator, and so its eigenstates constitute a full basis in the Hilbert space of the wave function states...

Hey! :giggle: Let $1\leq n\in \mathbb{N}$ and for $x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}, \ x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\in \mathbb{R}^n$ and let $x\cdot y=\sum_{i=1}^nx_iy_i$ the dot product of $x$ and $y$. Let $S=\{v\in \mathbb{R}^n\mid v\cdot...

The reference definition and problem statement are shown below with my work shown following right after. I would like to know if I am approaching this correctly, and if not, could guidance be provided? Not very sure. I'm not proficient at formatting equations, so I'm providing snippets, my...

Good afternoon to all again! I'm solving last year's problems and can't cope with this problem:( help me to understand the problem and find a solution!

Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an...

Hello, Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There...

We usually talk about ##F[[x]]##, the set of formal power series with coefficients in ##F##, as a topological ring. But we can also view it as a topological vector space over ##F## where ##F## is endowed with the discrete topology. And viewed in this way, ##\{x^n:n\in\mathbb{N}\}## is a...

Problem: Let ## V ## be a vector space over ## \mathbb{F} ## and suppose its dimension is even, ## dimV=2k ##. Show there exists an isomorphism ## \phi:V→V ## s.t. ## \phi(\phi(v))=−v ## for all ## v \in V ## Generally that way to solve this is to define a basis for the vector space ## V ##...

Hey! :giggle: We have the following linear maps \begin{align*}\phi_1:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}x+y\\ x-y\end{pmatrix} \\ \phi_2:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto...