Basis Definition and 1000 Threads

I am given the following set of 4x4 matrices. How can i justify that they form a basis for the Lie Algebra of the group SO(4)? I know that they must be real matrices, and AA^{T}=\mathbb{I}, and the detA = +-1. Do i show that the matrices are linearly independent, verify these properties, and...

The Kraus operator is defined as, $$A_{k}(t)={\sum_{\{k_i\}}^{k}}'\langle\{k_i\}|U(t)|\{0\}\rangle$$ is given in eqn(5) in the [Arxiv link](https://arxiv.org/pdf/quant-ph/0407263.pdf) the matrix representation of $A_k(t)$ is given in eqn (7) as...

Hey! :o Let $U_1,U_2$ be vector subspaces of $\mathbb{R}^4$ that are defined as $$U_1=\begin{bmatrix} \begin{pmatrix} 3\\ 2\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 3\\ 3\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 2\\ 1\\ 2\\ 1 \end{pmatrix} \end{bmatrix}, \ \ U_2=\begin{bmatrix}...

Is it correct, at least in the context of general relativity, to say that in a coordinate basis, the inner product between space-like basis vectors will be 1, and in a non-coordinate basis the inner product will be defined by the corresponding component of the metric? Can I take this conditions...

For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...

Hey! :o Let $t\in \mathbb{R}$ and the vectors $$v_1=\begin{pmatrix} 0\\ 1\\ -1\\ 1 \end{pmatrix}, v_2=\begin{pmatrix} t\\ 2\\ 0\\ 1 \end{pmatrix}, v_3=\begin{pmatrix} 2\\ 2\\ 2\\ 0 \end{pmatrix}$$ in $\mathbb{R}^4$. I want to determine a maximal linearly independent subset of $\{v_1...

Homework Statement Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3 = (0,1,-1)## Homework EquationsThe Attempt at a Solution I was thinking that, it looks like the matrix would have to...

So I am in an introductory modern physics class and we discussed how intrinsic spin can be a linear combination of the spin basis. I am a bit confused on the physical representation of this and whether or not there are different basis to represent spin. If it is possible, what would be the point...

Hey! :o I want to prove that $$V=\left \{\begin{pmatrix}a & b\\ c & d\end{pmatrix} \mid a,b,c,d\in \mathbb{C} \text{ and } a+d\in \mathbb{R}\right \}$$ is a $\mathbb{R}$-vector space. I want to find also a basis of $V$ as a $\mathbb{R}$-vector space. We have the following: Let $K$ be a field...

Is there a set of relationships for the wedge product of basis vectors as there are for the dot product and the cross product? i.e. e1*e1 = 1 e1*e2 = 0 e1 x e2 = e3

Hi, Excuse me this is probably a really stupid question but I ask because I thought that the definition of the dimension of a space is the number of elements in the basis. Now I have a theorem that tells me that ## dim M_{k} = [k/12] + 1 if k\neq 2 (mod 12) =[k/12] if k=2 (mod 12) ## for ## k...

Hey! :o We are given the vectors $\vec{a}=\begin{pmatrix}4\\ 1 \\ 0\end{pmatrix}, \vec{b}=\begin{pmatrix}2\\ 0 \\ 1\end{pmatrix}, \vec{c}=\begin{pmatrix}0\\ -2 \\ 4\end{pmatrix}$. I have shown by calculating the deteminant $|D|=0$ that these three vectors are linearly dependent. I want to...

I know that to find the projection of an element in R^n on a subspace W, we need to have an orthogonal basis in W, and then applying the formula formula for projections. However, I don;t understand why we must have an orthogonal basis in W in order to calculate the projection of another vector...

I'm a little bit confused about how coordinate systems work once we have chosen a basis for a vector space. Let's take R^2 for example. It is known that if we write a vector in R^2 numerically, it must always be with respect to some basis. So the vector [1, 2] represents the point (1, 2) in the...

Homework Statement T/F: If a finite set of vectors spans a vector space, then some subset of the vectors is a basis. Homework EquationsThe Attempt at a Solution It seems that the answer is true, due to the "Spanning Set Theorem," which says that we are allowed to remove vectors in a spanning...

Hi, I am struggling with the following problem: "Let $V=P_3(\Bbb{R})$ and let $t_1=3x^3-x-2$ and $t_2=x^3-3x+2$ with $T=\left\{ t\in V \:|\: t(1)=0 \right\}$. Find ${t_3}\in\left\{T\right\}$ such that $\left\{t_1, t_2, t_2\right\}$ is a basis of T. Not sure where to go as each column matrix...

To me it seems basic question or even obvious but as I am not mathematician I would rather like to check. Is it true that these two matrices are both identity matrices: ##\begin{pmatrix}1&0\\0&1\end{pmatrix} ## and...

Homework Statement ##\mathbb{H} = \{(a,b,c) : a - 3b + c = 0,~b - 2c = 0,~2b - c = 0 \}## Homework EquationsThe Attempt at a Solution This definition of a subspace gives us the vector ##(3b - c,~2c,~2b) = b(3,0,2) + c(-1,2,0)##. This seems to suggest that a basis is {(3, 0, 2), (-1, 2 0)}, and...

I have the following question: Is there a basis for the vector space of polynomials of degree 2 or less consisting of three polynomial vectors ##\{ p_1, p_2, p_3 \}##, where none is a polynomial of degree 1? We know that the standard basis for the vector space is ##\{1, t, t^2\}##. However...

Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...

Homework Statement Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V. Homework Equations http://linear.ups.edu/html/section-VS.html We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly...

In most situations in QM we would get a quantized energy basis, that is a countably infinite basis ( I think it's called having a cardinality of aleph 0), In the meanwhile we take the position basis to be continuous ( cardinality of aleph 1?) and I'm pretty sure that there is a theorem stating...

I have two n-vectors e_1, e_2 which span a 2D subspace of R^n: V = span\{e_1,e_2\} The vectors e_1,e_2 are not necessarily orthogonal (but they are not parallel so we know its a 2D and not a 1D subspace). Now I also have a linear map: f: V \rightarrow W \\ f(v) = A v where A is a given n...

To determine the mass of charged leptons, we rotate such that the matrix of yukawa couplings (which gives the mass matrix after EWSB) is diagonal. We also call this flavour basis for neutrinos, because the flavoured neutrinos couple directly to the correspondong flavoured lepton in weak charged...

Hello, can someone give an example for an incoherent State --> a formula is here on page 7 : http://quantumcorrelations.weebly.com/uploads/6/6/5/5/6655648/2016_robustnessofcoherencetalk.pdf I know that coherenc is e.g. a superposition of e.g. Spin-Up and Spin-Down [z] or so... But i have no...

I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...

Dear all, The Hamiltonian for a particle in a magnetic field can be written as $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$ where ##\boldsymbol\sigma## are the Pauli matrices. This Hamiltonian is written in the basis of the eigenstates of ##\sigma_z##, but how is it...

Homework Statement A spin-1/2 electron in a magnetic field can be regarded as a qubit with Hamiltonian $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$. This matrix can be written in the form of a qubit matrix $$ \begin{pmatrix} \frac{1}{2}\epsilon & t\\ t^* &...

Hi all My question: One of the four Bell's state are measuring in the Bell basis. Whether the result of measurement of this one of the four Bell's state will be the same Bell's state (just that Bell's state which are measuring) ? The each of four Bell's state is a quantum superposition of the...

Hi, I have recently studied about basis for wavelet function which is helpful to design any function. Likewise, what is the basis for bessel function and how can it be implemented for an image ( because image is also a function). Specifically, I am interested to know how bessel function can be...

Could you view a discrete number, for instance a binary number, as a sort of orthogonal basis, where each digit position represents a new dimension? I see similarities between a binary number and for instance Fourier Transform, with each digit being a discrete function.

So I am currently learning some regression techniques for my research and have been reading a text that describes linear regression in terms of basis functions. I got linear basis functions down and no exactly how to get there because I saw this a lot in my undergrad basically, in matrix...

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...

Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...

Let A = (a_{ij}) be a k\times n matrix of rank k . The k row vectors, a_i are linearly independent and span a k-dimensional plane in \mathbb{R}^n . In "Geometry, Topology, and Physics" (Ex 5.5 about the Grassmann manifold), the author states that for a matrix g\in...

Kramer's Law describes the spectrum of Bremsstrahlung: What is the theoretical basis as to why the spectrum assumes this shape?

Hello, I am trying to optimize a molecule with the chemical formula C60H52O18P4S4W2. I have tried using different Basis Sets on my input file but I keep getting the following error: " Standard basis: Aug-CC-pV5Z (5D, 7F) Atomic number out of range in CCPV5Z. Error termination via Lnk1e in...

Homework Statement In Euclidean three-space, let ##p## be the point with coordinates ##(x,y,z)=(1,0,-1)##. Consider the following curves that pass through ##p##: ##x^{i}(\lambda)=(\lambda , (\lambda -1)^{2}, -\lambda)## ##x^{i}(\mu)=(\text{cos}\ \mu , \text{sin}\ \mu , \mu - 1)##...

Homework Statement I want to find the matrix representation of the ##\hat{S}_x,\hat{S}_y,\hat{S}_z## and ##\hat{S}^2## operators in the ##S_x## basis (is it more correct to say the ##x## basis, ##S_x## basis or the ##\hat{S}_x## basis?). Homework Equations...

I have a function of a 3 vector, i.e. f(+x,+y,+z) [ or for conveniance f=+++] this function is repeated 4 times where: f1 = + + + f2 = + - + f3 = - - + f4 = - + + I need a formula where i have a different vector for each function in a summation, to obtain the superposition of all 4...

Why are basis vectors represented with subscripts instead of superscripts? Aren’t they vectors too? Isn’t a vector a linear combination of basis vectors (and not basis co-vectors?) In David McMahon’s Relativity Demystified, he says, “We will often label basis vectors with the notation e_a...

Homework Statement Write down an orthonormal basis of 1 forms for the rotating C-metric [/B] Use the result to find the corresponding dual basis of vectorsSee attached file for metric and appropriate equations The two equations on the left are for our vectors. the equations on the right...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows: I do not...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...

Hello all! I've just started to study general relativity and I'm a bit confused about dual basis vectors. If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...

Homework Statement Please see the attached picture Homework Equations Reduced echelon form of the column matrix The Attempt at a Solution I can solve for the first part to find which ones are the bases in ##\mathbb{R}^3## by determining whether in the echelon form, there is a pivot in each...

1. Homework Statement I've found the dimension of V to be 3. According to the solutions, it seems that the basis can be written straight away, { (1,1,1,2), (1,2,-3,1), (3,4,-1,5) } (which is also the basis for the column space of the matrix), without verifying the vectors are linearly...

Hey could anyone please explain how you go about drawing a reciprocal lattice? For example a 2d rectangular lattice to it's reciprocal form? Also... I don't know if this is correct but if you have a 2d rectangular lattice with lattice vectors L=n1a1 + n2a2 would the reciprocal lattice vectors...