Basis Definition and 1000 Threads

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. Rick16

    I Dot product between basis vectors and dual basis vectors

    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...
  2. S

    I Which basis-forms are pseudo-tensors?

    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##...
  3. F

    I Non orthogonal basis and the lines of its coordinate grid

    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...
  4. F

    I Standard basis and other bases...

    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...
  5. mattTch

    I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

    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).
  6. S

    Find a basis for W which is subset of V

    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...
  7. S

    I Can time be another basis vector under Galilean relativity?

    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...
  8. S

    I Orthonormal basis expression for ordinary contraction of a tensor

    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...
  9. S

    I Consistent matrix index notation when dealing with change of basis

    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...
  10. James1238765

    I Use of Gell-Mann matrices as the SU(3) basis for gluon states?

    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...
  11. P

    A Proof of the inequality of a reduced basis

    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...
  12. S

    I Operators in finite dimension Hilbert space

    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 , |...
  13. Euge

    POTW A Modified Basis in an Inner Product Space

    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##.
  14. K

    A Matrix representation of a unitary operator, change of basis

    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...
  15. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

    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...
  16. H

    Vector space of functions defined by a condition

    ##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##...
  17. C

    A FEM basis polynomial order and the differential equation order

    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}...
  18. theycallmevirgo

    What is the basis for Toyota's fluorine battery claims?

    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
  19. LCSphysicist

    Covariant derivative in coordinate basis

    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...
  20. cianfa72

    I Can a Basis Vector be Lightlike?

    [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...
  21. P

    A Position basis in Quantum Mechanics

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

    I Finding the orthogonal projection of a vector without an orthogonal basis

    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...
  23. G

    B Black Hole Entropy: Basis of Logarithm Explored

    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...
  24. H

    Find the basis so that the matrix will be diagonal

    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...
  25. yucheng

    Unit Basis Components of a Vector in Tensorial Expressions?

    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}}...
  26. L

    I Basis of 2x2 matrices with real entries

    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...
  27. e2m2a

    I Confusion between vector components, basis vectors, and scalars

    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...
  28. H

    A quite verbal proof that if V is finite dimensional then S is also....

    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 =...
  29. Z

    I "Approximating Basis" -- Is there a contemporary term?

    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...
  30. H

    I Is This a Valid Basis for the Set of Polynomials with \( p(0) = p(1) \)?

    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...
  31. H

    I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?

    ##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...
  32. A

    Check that the polynomials form a basis of R3[x]

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

    I Similarity transformation, basis change and orthogonality

    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...
  34. P

    I Measurement of a qubit in the computational basis - Phase estimation

    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...
  35. J

    MHB Finding a Basis for a Linear Subspace Orthogonal to a Given Point P in R^3

    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...
  36. evinda

    MHB Converting Numbers Between Different Bases: Is It Possible to Use Only n Digits?

    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 $n$...
  37. George Keeling

    LaTeX Plane polar noncoordinate basis (Latex fixed)

    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},\...
  38. A

    I Expressing Vectors of Dual Basis w/Metric Tensor

    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##...
  39. BigMax

    I The Meaning of Basis States in Quantum Mechanics

    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...
  40. M

    B Exploring Holonomic Basis in Cartesian Coordinates

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

    A Does the quantum space of states have countable or uncountable basis?

    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...
  42. M

    MHB Diagonalizable transformation - Existence of basis

    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...
  43. A

    Riesz Basis Problem: Definition & Problem Statement

    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...
  44. W

    MHB Find Eigenvalues & Basis C2 Matrix: Help!

    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!
  45. L

    A What Topological Vector Spaces have an uncountable Schauder basis?

    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...
  46. F

    I Change of Basis Matrix vs Transformation matrix in the same basis....

    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...
  47. L

    A Can falling factorials be a Schauder basis for formal power series?

    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...
  48. C

    Existence of isomorphism ϕ:V→V s.t. ϕ(ϕ(v))=−v for all v∈V

    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 ##...
  49. M

    MHB Give a basis to get the specific matrix M

    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...
  50. Antarres

    A Transformation of coordinate basis

    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...
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