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. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  2. D

    Expectation Value of Q in orthonormal basis set Psi

    Homework Statement Suppose that { |ψ1>, |ψ2>,...,|ψn>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψj> = qj|ψj> for all j = 1...n. A particle is in the state |Φ>. Show that for this particle the expectation value of <Q> is ∑j=1nqj |<Φ|...
  3. Mentz114

    I Spin and polarization basis problem?

    It is well established that Fermionic or quantum spin is described by the Pauli matrices and their algebra which give the basis vectors and operators. Consider a superposition of spin-up and spin-down states ##\psi_S = \cos(\alpha)|z_+\rangle + \sin(\alpha)|z_-\rangle## and project into...
  4. RicardoMP

    I Diagonalization and change of basis

    I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle: \begin{bmatrix} E_0 &-A \\ -A & E_0 \end{bmatrix} Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}}\\...
  5. L

    MHB Orthonormal Basis times a real Matrix

    Hi! I have an orthonormal basis for vector space $V$, $\{u_1, u_2, ..., u_n\}$. If $(v_1, v_2, ..., v_n) = (u_1, u_2, ... u_n)A$ where $A$ is a real $n\times n$ matrix, how do I prove that $(v_1, v_2, ... v_n)$ is an orthonormal basis if and only if $A$ is an orthogonal matrix? Thanks!
  6. LarryS

    I Measures of Linear Independence?

    My formal education in Linear Algebra was lacking, so I have been studying that subject lately, especially the subject of Linear Independence. I'm looking for functions that would qualify as measures of linear independence. Specifically, given a real-valued vector space V of finite dimension...
  7. I like Serena

    MHB What is the basis of the trivial vector space {0}

    It makes me wonder... wikipedia says about a basis: In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.[1] So what is the...
  8. M

    Find largest number of linearly dependent vectors among these 6 vectors

    Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...
  9. N

    B Tensor Product, Basis Vectors and Tensor Components

    I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
  10. A

    Components of a vectors with respect to basis are unique?

    Mentor note: Moved from Intro Physics, as this is more of a mathematics question Homework Statement With respect to a prescribed basis... |e 1> |e 2> ...|e n> Any given vector |a> = a1|e1> +a2|e2>+...+a n|e n>, Is uniquely represented by the (order) n-ruble of its components. |a> <--->(...
  11. R

    Can a Scalar Multiple of a Basis Still Form a Basis?

    Homework Statement I'm using this problem, and although I got the correct answer my question is really about the properties of a basis, not the problem that I'm going to state.. anyways, Let = {v_1, v_2, ..., v_n } be a basis for a vector space V. Let c be a nonzero scalar. Show that the set...
  12. R

    Find a basis for R2 that includes the vector (1,2)

    My answer was [ (1,2), (0,-1) ] I just wanted to make sure with you guys that you could essentially have an infinite amount of answers. For example: [ (1,2), (0,1) ] [ (1,2), (0, -5) ] etc would all suffice, right?
  13. S

    Find the basis of a vector subspace of R^2,2

    Homework Statement i know how to find the basis of a subspace of R2 or R3 but I can't figure out how to find the basis of a subspace of something like R2,2. I even have an example in my book which i managed to follow nearly till the end but not quite... Given matrix: A= 6 -9 4 -4 show that...
  14. FallenApple

    A Infinite Basis and Supernatural Numbers

    I've read that it is unsatisfactory to consider infinitely many basis vectors to span an infinite dimensional space. For example, for the infinite dimensional Hilbert space, {e1,e2,e3...} we could use this to make an arbitrary infinite tuple (a,b,c,...). If this is looked down upon, then why are...
  15. N

    Topological Basis Homework: If-Then Conditions

    Homework Statement Let (χ,τ) be a topological space and β be a collection of subsets of χ. Then β is a basis for τ if and only if: 1. β ⊂ τ 2. for each set U in τ and point p in U there is a set V in β such that p ∈ V ⊂ U. 2. Relevant definitions Let τ be a topology on a set χ and let β ⊂ τ...
  16. Adrian555

    Natural basis and dual basis of a circular paraboloid

    Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by: $$x = \sqrt U cos(V)$$ $$y = \sqrt U sen(V)$$ $$z = U$$ with the inverse relationship: $$V = \arctan \frac{y}{x}$$ $$U = z$$ The natural basis is: $$e_U = \frac{\partial \overrightarrow{r}}...
  17. J

    A Quantum measurement with incomplete basis?

    In page 9 of http://www.theory.caltech.edu/people/preskill/ph229/notes/chap4.pdf we can find simple form of Bell inequalities for three binary variables: $$ P(A=B) + P(A=C) + P(B=C) \geq 1 $$ which is kind of obvious: "This is satisfied by any probability distribution for the three coins because...
  18. JTC

    A Understanding the Dual Basis and Its Directions

    Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing...
  19. Eclair_de_XII

    Finding a basis for the linear transformation S(A)=A^T?

    Homework Statement "Find ##S_\alpha## where ##S: M_{2×2}(ℝ)→M_{2×2}(ℝ)## is defined by ##S(A)=A^T##. Homework Equations ##A^T=\begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix}## ##\alpha= \{ {\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0...
  20. Ron19932017

    I Self-Study GR: Construct Contravarient/Covarient Orthogonal Basis

    Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...
  21. Elroy

    Linear Algebra Problem: Solving for Euler between two ordered bases

    Homework Statement Linear Algebra Problem: Solving for Euler between two ordered bases I've got a problem I need to solve, but I can't find a clean solution. Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...
  22. G

    What Determines Voltage Reference and Measurement Accuracy?

    An example of how trying to teach someone exposes gaps in your knowledge: I was talking to someone about why we have line, neutral and Earth conductors. We got on to the fact that the Earth is an enormous sink and source of charge, so anything referenced to it can be considered 0 volts. Then...
  23. C

    I Coordinate system vs ordered basis

    I have an issue with the definition of coordinate system in differential geometry vs the definition of coordinate system in linear algebra. The post is a bit long, but it's necessary so that I get my point across. Let ##V## be an ##n##-dimensional normed space over the reals and equip ##V##...
  24. AJ007

    I 2 basis sets, for which one is DFT calc faster?

    Hello fellow physicist, I'm new on this subject, I hope that you can help me clear some stuff out. If I have 2 basis sets; A containing 100 basis functions and B containing 50 basis functions, for which one of them will a DFT calculation be faster? Also, is a DFT calculation using basis sets...
  25. P

    I Differential forms as a basis for covariant antisym. tensors

    In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
  26. Vitani11

    Representation of vectors in a new basis using Dirac notation?

    Homework Statement I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that |v⟩ =...
  27. A

    Proving Completeness of Continuous Basis Vectors

    Homework Statement Consider the vector space that consists of all possible linear combinations of the following functions: $$1, sin (x), cos (x), (sin (x))^{2}, (cos x)^{2}, sin (2x), cos (2x)$$ What is the dimension of this space? Exhibit a possible set of basis vectors, and demonstrate that...
  28. T

    I Hermitian operators, matrices and basis

    Hello, I would just like some help clearing up some pretty basic things about hermitian operators and matricies. I am aware that operators can be represented by matricies. And I think I am right in saying that depending on the basis used the matrices will look different, but all our valid...
  29. M

    I Understanding Change of Basis & Superpositioning of States

    I think I do not quite understand the role that change of basis plays is superpositioning of states. If there is an observable, ##A## which is represented by the operator ##\hat{A}##, then the set of observed values for that observable will be the set of eigenvalues defined by the operator...
  30. Destroxia

    Pauli Matrices in the Basis of Y?

    Homework Statement [/B] I know the pauli matrices in terms of the z-basis, but can't find them in terms of the other bases. I would like to know what they are. Homework Equations The book says they are cyclic, via the relations XY=iZ, but this doesn't seem to apply when I use this to find the...
  31. davidge

    I Constructing Vector in Another Basis: A Guide

    Suppose we have defined a vector ##V## at a point ##x##, so it has components ##V^\mu(x)## at ##x##. Let ##y## be another point, such that ##y^\mu = x^\mu + \epsilon \zeta^\mu(x)##, ##\epsilon## a scalar. Now, since ##x## and ##y## are coordinate points, the vector ##V## should not depend on...
  32. Minal

    A Find vectors in Orthogonal basis set spanning R4

    An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4. If v1 and v2 are [ −1 2 3 0 ] and [−1 1 −1 0 ] find v3 and v4. Please explain this in a very simple way.
  33. M

    Various Properties of Space Curve....*Really *

    Homework Statement Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'. (i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc...
  34. Poetria

    Basis for the space of solutions (ODE)

    Homework Statement The equation given: dy/dt = 3*y A basis for the space of solutions is required.The Attempt at a Solution According to me it is e^(3*t) but it has turned out false. Why? I am considering the answer "The basis is the set of all functions of the form c*e^(3*t) but a...
  35. nacreous

    Calculate coefficients of expansion for vector y

    Homework Statement Let v(0) = [0.5 0.5 0.5 0.5]T, v(1) = [0.5 0.5 -0.5 -0.5]T, v(2) = [0.5 -0.5 0.5 -0.5]T, and z = [-0.5 0.5 0.5 1.5]T. a) How many v(3) can we find to make {v(0), v(1), v(2), v(3)} a fully orthogonal basis? b) What are z's coefficients of expansion αk in the basis found in...
  36. L

    I What is the Basis for the Null Space in Matrix A?

    Hello there. I'm currently trying to come to terms with the aforementioned topics. As I am self studying, a full understanding of these concepts escapes me. There's something I'm not grasping here and I would like to discuss these to clear away the clouds. As I understand it, a basis for some...
  37. bananabandana

    Why are unperturbed states valid basis for perturbed system?

    Homework Statement So we have a two state system, with unperturbed eigenstates ## |\phi_{1}\rangle##, ## |\phi_{2}\rangle ##, and Hamiltonian ## \mathbf{\hat{H_{0}}}## - i.e ##\mathbf{\hat{H_{0}}}|\phi_{1}\rangle = E_{1}|\phi_{1}\rangle## We shine some z-polarized light on the system. This...
  38. 0

    Eigenvectors and orthogonal basis

    Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
  39. D

    I Change of Basis and Unitary Transformations

    Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A. Now, the first (old) basis, in terms of the second(new) basis, is given by, say, $$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$ How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for...
  40. M

    Preferred Basis solution in MWI

    For the difficulty of getting a preferred basis in the unitary only MWI. I saw this message by Fredrik: "If we just let it go, it seems very natural to me to (if we insist on trying to interpret QM as a description of the universe) postulate something like "every 1-dimensional subspace of the...
  41. Zero2Infinity

    Basis of the intersection of two spaces

    Homework Statement Consider two vector spaces ##A=span\{(1,1,0),(0,2,0)\}## and ##B=\{(x,y,z)\in\mathbb{R}^3 s.t. x-y=0\}##. Find a basis of ##A\cap B##. I get the solution but I also inferred it without all the calculations. Is my reasoning correct Homework Equations linear dependence...
  42. T

    I Is the trace of a matrix independent of basis?

    Hello, Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of. Thank you
  43. M

    MHB Degree of transcendental basis

    Hey! :o Let $E_1, E_2$ be extensions of the field $F$, that are contained in a bigger field, so it makes sense to consider the field $E_1E_2$. Without using the proposition: $F \leq K \leq E \Rightarrow \text{tr.deg}(E/F) = \text{tr.deg}(E/K) + \text{tr.deg}(K/F)$ I want to show that...
  44. G

    I Quantum Superposition, Linear Combinations and Basis

    Hello, just a quick question. I am aware that a a state in a space can be written as a linear combination of the basis kets of that space ψ = ∑ai[ψi] where ai are coefficients and [ψi] are the basis vectors. I was just wondering is this a linear superposition of states or just a linear...
  45. Matejxx1

    Find the basis of a kernel and the dimension of the image

    Homework Statement Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)## a)Proof that this map is linear b)Find some basis of the kernel b)Find the dimension of the image Homework Equations ##\mathbb{R}_{n}[x]##...
  46. M

    Proving Linear Independence of Av1, ..., Avk and Conditions for Basis in Rm

    Homework Statement [/B] 1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}. (a) Prove that {Av1, . . . , Avk} is linearly independent. (b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...
  47. L

    B Help Me Understand Rotation from an Arbitrary basis

    Apologies on the vague title, I'm having some trouble describing what I'm trying to do and need some context. I have a 3D World Coordinate system that is static, using an extrinsic left-handed ZYX system. I have an object with a radius of 1 unit (generic), this is arbitrarily placed within the...
  48. F

    I Basis of a Subspace of a Vector Space

    Hello Forum and happy new year, Aside from a rigorous definitions, a linear vector space contains an infinity of elements called vectors that must obey certain rules. Based on the dimension ##N## (finite or infinite) of the vector space, we can always find a set of ##n=N## linearly independent...
  49. O

    A Justify matrices form basis for SO(4)

    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...
  50. Muthumanimaran

    A Kraus Operator in Fock basis

    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...
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