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 Determining Basis and Dimension for Vector Subspaces $U_1$ and $U_2$

    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}...
  2. K

    I Coordinate vs. Non-Coordinate Basis in General Relativity

    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...
  3. T

    A Are all wave functions with a continuum basis non-normalizable?

    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...
  4. M

    MHB Extend the vectors to a 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...
  5. Mr Davis 97

    Construct a differential equation from the basis of solution

    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...
  6. yeezyseason3

    I Understanding Spin Basis & Representation in Modern Physics | Explained

    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...
  7. M

    MHB No prefix added: How do I find a basis of a $\mathbb{R}$-vector space?

    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...
  8. Kevin McHugh

    I Wedge product of basis vectors

    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
  9. binbagsss

    A Modular forms, dimension and basis confusion, weight mod 1

    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...
  10. M

    MHB Is there a more efficient way to determine the basis of a subspace?

    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...
  11. Mr Davis 97

    I Orthogonal basis to find projection onto a subspace

    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...
  12. Mr Davis 97

    I Representing vectors with respect to a basis

    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...
  13. Mr Davis 97

    T/F: Subset of a spanning set always forms a basis

    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...
  14. H

    MHB Finding a 3rd polynomial to create a basis.

    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...
  15. zonde

    B Is identity matrix basis dependent?

    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...
  16. Mr Davis 97

    Dimension and basis for a subspace

    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...
  17. Mr Davis 97

    I Existence of basis for P_2 with no polynomial of degree 1

    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...
  18. M

    I Proof that every basis has the same cardinality

    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...
  19. K

    Proving That Any Vector in a Vector Space V Can Be Written as a Linear Combination of a Basis Set

    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...
  20. N

    I Should the position basis be quantized?

    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...
  21. V

    I How to Construct an Orthonormal Basis for a 2D Subspace in Linear Algebra?

    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...
  22. V

    A Transforming lepton basis to diagonlise charged lepton mass

    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...
  23. P

    I States diagonal in the reference basis

    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...
  24. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

    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...
  25. I

    I How to change the Hamiltonian in a change of basis

    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...
  26. I

    Magnetic field Hamiltonian in different basis

    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^* &...
  27. L

    B Bell's state measurement in the Bell basis

    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...
  28. G

    What is the basis for bessel function as we have for wavelet

    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...
  29. entropy1

    B Can a Binary Number be Viewed as an Orthogonal Basis?

    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.
  30. J

    A Linear Regression with Non Linear Basis Functions

    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...
  31. chi_rho

    A Transforming Spin Matrices (Sx, Sy, Sz) to a Spherical Basis

    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...
  32. P

    Linear transformation representation with a matrix

    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...
  33. J

    I Multiplication by a matrix in GL rotates a plane's basis?

    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...
  34. T

    I What is the theoretical basis for Kramer's Law?

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

    A Why does my chosen basis set for W crash (DFT)?

    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...
  36. S

    Tangent vectors in the coordinate basis

    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)##...
  37. P

    Representing spin operators in alternate basis

    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...
  38. Ben Wilson

    I What are the necessary trig functions for finding the rotation formula?

    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...
  39. snoopies622

    I Confused about basis vector notation

    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...
  40. L

    Orthonormal basis of 1 forms for the rotating c metric

    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...
  41. Math Amateur

    I Basis of a Tensor Product - Theorem 10.2 - Another Question

    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...
  42. Math Amateur

    I Basis of a Tensor Product - Cooperstein - Theorem 10.2

    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...
  43. Math Amateur

    MHB Basis of a Tensor Product - Theorem 10.2 - Another Question .... ....

    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...
  44. Math Amateur

    MHB Basis of a Tensor Product - Cooperstein - Theorem 10.2

    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...
  45. V

    I Confusion about Dual Basis Vectors: Why are these two relationships equal?

    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...
  46. M

    Determine which system of vectors span C^3

    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...
  47. J

    Finding a basis for the range space

    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...
  48. thegirl

    I Drawing a reciprocal lattice, also basis

    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...
  49. Math Amateur

    MHB Dual Vector Space and Dual Basis - another question - Winitzki Section 1-6

    I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.6: Dual (conjugate) vector space ... ... I need help in order to get a clear understanding of an aspect of the notion or concept of the dual basis \{ e^*_1, e^*_2, \ ... \ ... \...
  50. G

    MHB Change of Basis: Exploring Basis Vectors in $\mathbb{R}^3$

    Consider the following set of vectors in $\mathbb{R}^3:$ $u_0 = (1,2,0),~ u_1 = (1,2,1), ~u_2 = (2,3,0), ~u_3 = (4,6,1)$ Explain why each of the two subsets $B_0 = \left\{u_0, u_2,u_3\right\}$ and $B_1 = \left\{u_1, u_2, u_3\right\}$ forms a basis of $\mathbb{R}^3$. If we write $[\mathbf{x}]_0$...
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