What is Basis: Definition and 1000 Discussions

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).

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  1. M

    MHB Orthonormal basis for the poynomials of degree maximum 2

    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 \\...
  2. Supantho Raxit

    A What is a good basis for coupled modes in a resonator?

    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...
  3. P

    Legendre Polynomials as an Orthogonal Basis

    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...
  4. BWV

    A Integration with respect to a Lévy basis / Ambit fields

    Familiar with basics of stochastic calculus and integration over a Brownian motion. Trying to get a sense of Ambit Fields https://en.wikipedia.org/wiki/Ambit_field which mention an integration over a Lévy basis: Curious if anyone familiar with this? A Brownian motion is a Levy process...
  5. penroseandpaper

    I Showing a set is a basis for a vector space

    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...
  6. K

    Linear algebra, find a basis for the quotient space

    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...
  7. E

    B Just a question about the tangent basis

    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...
  8. K

    MHB Determine the area, calculate the basis vectors and determine the inner product

    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 defined. Express the...
  9. K

    Calculate the dual basis and tangent basis vectors

    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...
  10. K

    I Understanding the concepts of isometric basis and musical isomorphism

    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)...
  11. Athenian

    Finding the Basis Vectors for a Coordinate System

    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...
  12. R

    I Measuring Entangled Particles in two different Basis

    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 =...
  13. S

    I Why should a Fourier transform not be a change of basis?

    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...
  14. L

    A Topological Basis in l2 Space: Why?

    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?
  15. S

    B Relating basis vectors at different points in a neighborhood

    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...
  16. LCSphysicist

    Basis and positive basis

    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...
  17. George Keeling

    A Exploring Null Basis Vectors, Metric Signatures Near Kruskal

    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...
  18. LCSphysicist

    Changing Basis: Matrix E->F vs. Matrix B->C

    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?
  19. S

    B Reconciling basis vector operators with partial derivative operators

    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...
  20. E

    B Do column 'vectors' need a basis?

    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}##...
  21. S

    I Tangent space basis vectors under a coordinate change

    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...
  22. J

    I Metric defined with a non-coordinate basis

    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##...
  23. R

    I The Levi-Civita Symbol and its Applications in Vector Operations

    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 =...
  24. M

    MHB Orthonormal basis - Set of all isometries

    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...
  25. D

    MHB Vector of polynomial and basis.

    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...
  26. Eth338

    I What is the Relationship Between Lattice Points and Basis Balls?

    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...
  27. M

    MHB Show that there are vectors to get a basis

    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...
  28. L

    A Coherent states: Orthonormal set? Overcomplete basis?

    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...
  29. forkosh

    A Exploring Basis Vector Relationships in Incompatible Propositions

    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...
  30. Math Amateur

    I Local Basis in Topology .... Definitions by Croom and Singh .... ....

    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...
  31. Frigus

    On what basis are parts of the brain classified?

    On the basis of what brain is classified,like we say on the basis of function the neurons are of 3 types.
  32. Avaro667

    A Elliptic trigonometric functions as basis for function expansion ?

    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...
  33. E

    Why doesn't using a basis which is not orthogonal work?

    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...
  34. berlinspeed

    I Notation inquiry - bar over basis set....

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

    I Understanding the Derivation of Reciprocal Lattice Basis from Equations 5 and 6

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

    I Calculating group representation matrices from basis vector/function

    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...
  37. M

    MHB Proving $\{w_1, \ldots , w_m\}$ is a Basis of $\text{Lin}(v_1, \ldots , v_k)$

    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 $\text{Lin}(v_1, \ldots , v_k)$...
  38. M

    A Finding eigenvalues with spectral technique: basis functions fail

    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...
  39. whoohm

    Maple Anyone using Maple in 2019 on regular basis?

    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...
  40. P

    A Christoffel Symbols in terms of a Change in Basis

    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...
  41. Pencilvester

    I Lie derivative of hypersurface basis vectors along geodesic congruence

    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...
  42. S

    Understanding Eigenvectors: Solving for Eigenvalues and Corresponding Vectors

    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!
  43. S

    I How do you derive those basis vectors in GR?

    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...
  44. archaic

    B Isn't the concept of a basis circular?

    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...
  45. M

    A Vector space (no topology) basis

    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...
  46. T

    A How do I find the change of basis matrix for the JCF of M?

    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...
  47. J

    I Exploring the Role of Basis in Describing Objects and Reality in Physics

    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...
  48. karush

    MHB 14.3 Find a basis for NS(A) and dim{NS(A)}

    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]=...
  49. karush

    MHB 14.2 find a basis for NS(A) and dim{NS(A)}

    $\tiny{370.14.2.}$ For the matrix $A=\left[ \begin{array}{rrrr} 1&0&1\\0&1&3 \end{array}\right]$ find a basis for NS(A) and $\dim{NS(A)}$ ----------------------------------------------------------- altho it didn't say I assume the notation means Null Space of A Reducing the augmented matrix...
  50. karush

    MHB 13.2 verify that ....... is a basis for R^2 find [v]_beta

    Verify that $\beta=\left\{\begin{bmatrix} 0\\2 \end{bmatrix} ,\begin{bmatrix} 3\\1 \end{bmatrix}\right\}$ is a basis for $\Bbb{R}^2$ Then for $v=\left[ \begin{array}{c}6\\8\end{array} \right]$, find $[v]_\beta$ ok, I presume next is...
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