# 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. ### 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).

7. ### 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...
8. ### 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 , |...
9. ### 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##.
10. ### 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...
11. ### 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...
12. ### 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##...
13. ### 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}...
14. ### 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
15. ### 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...
16. ### 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...
17. ### 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?
18. ### 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...
19. ### 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...
20. ### 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. ### 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...
26. ### I Can this work as a basis for S?

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...
27. ### 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...
28. ### 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.
29. ### 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...
30. ### 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...
31. ### MHB Basis of linear subspace

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...
32. ### MHB Conversion to other basis

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 $n$...
33. ### 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},\...
34. ### 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##...
35. ### I The meaning of basis states

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...
36. ### B Exploring Holonomic Basis in Cartesian Coordinates

Are cartesian coordinates the only coordinates with a holonomic basis that's orthonormal everywhere?
37. ### 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...

50. ### 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...