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. mishima

    Write ∇u with covariant components and contravariant basis

    The first part I'm fairly sure is just the regular gradient in polar coordinates typically encountered: $$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$ or in terms of scale factors: $$=\sum \hat...
  2. The black vegetable

    I Orthonormal Basis - Definition & Examples

    Is this correct? If not any hints on how to find Many thanks
  3. PCal

    Composition of flue gases by volume on a wet basis and dry basis

    Wet basis 0.75mol C4H10 Requires 4.875 mols O2 Produces 3 mols of CO2 and 3.75 mols of H2O 0.1mol C3H8 Requires 0.5 mols O2 Produces 0.3 mols of CO2 and 0.4 mols of H2O 0.15mol C4H8 Requires 0.9 mols O2 Produces 0.6 mols of CO2 and 0.6 mols of H2O Theoretical oxygen= 6.3mol +10% excess...
  4. L

    I What is the Basis of a Composite System?

    If I have a composite system, like a two particle system, for exemple, I can construct my Hilbert space as the tensor product of the hilbert spaces of these particles, and, if ##\{|A;m \rangle \}## and ##\{|B;n \rangle \}## are basis in these hilbert spaces, a basis in the total hilbert space is...
  5. V

    Finding a basis for a subspace

    I had assumed that we had to put our values into a matrix so I did [1 2 -1 0; 1 -5 0 -1] and then I would do a=[1; 1] and repeat for b, c, and d. This is incorrect however. I also thought that it could be {(1, 2, -1, 0),(1, -5, 0, -1)} however this was not the answer, and I am unsure of what do...
  6. karush

    MHB -412.12.1 basis of kernel

    ok I am new to this basis of kernel and tried to understand some other posts on this but they were not 101 enough Find the basis for kernel of the differential operator $D:C^\infty\rightarrow C^\infty$, $D^4-2D^3-3D^2$ this can be factored into $D^2(D-3)(D+1)$
  7. M

    A Basis functions and spanning a solution space

    Hi PF Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation $$\langle...
  8. karush

    MHB 11.3 Give the matrix in standard basis

    We define the application $T:P_2\rightarrow P_2$ by $$T(p)=(x^2+1)p''(x)-xp'(x)+2p'(x)$$ 1. Give the matrix $\displaystyle\left[T\right]_\infty^\infty$ in the standard basis $\alpha=(x^2,x,1)$ 2 Give the matrix $\displaystyle\left[T\right]_\infty^\infty$ where...
  9. J

    MHB Showing the Dual Basis is a basis

    I am working through a book with my professor and we read a section on the dual space, $V^*$. It gives the basis dual to the basis of $V$ and proves that this is in fact a basis for $V^*$. Characterized by $\alpha^i(e_j)=\delta_j^i$ I understand the proof given. But he said a different...
  10. karush

    MHB Null Space of A: Find Rank & Dim.

    Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...
  11. karush

    MHB Verifying $\beta$ as a Basis of $\Bbb{R}^2$ & Finding $|v|_{\beta-}$

    verify that $\beta =\left\{ \left[\begin{array}{c}0 \\ 2 \end{array}\right], \left[\begin{array}{c}3 \\ 1\end{array}\right]\right\}$ is a basis for $\Bbb{R}^2$ for $v=\left[\begin{array}{c}6\\ 8 \end{array}\right]$ find $|v|_{\beta-}$ ok $x_2=2$ and $x_1=3$ not sure how to answer the rest
  12. P

    I Integration of the Outer Product of a Basis

    Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...
  13. Physics Learner

    Second Quantized Minimal Basis Hamiltonian of H2

    Hi, I am really new in understanding second quantization formalism. Recently I am reading this journal: https://dash.harvard.edu/bitstream/handle/1/8403540/Simulation_of_Electronic_Structure.pdf?sequence=1&isAllowed=y In brief, the molecular Hamiltonian is written as $$\mathcal{H}=\sum_{ij}...
  14. A

    I Tangent vector basis and basis of coordinate chart

    I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...
  15. V

    MHB Can Vector Space $(V,O_1,O_2)$ Represent 2 Graphs?

    Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...
  16. microsansfil

    I Does quantum entanglement depend on the chosen basis?

    Hi, In this presentation about quantum optics it is mentioned that the same quantum state |Ψ> has different expressions in different mode bases : factorized state or entangled state. This presentation is related to this video : In some way entanglement isn't intrinsic. It depend on the...
  17. A

    A Representing harmonic oscillator potential operator in. Cartesian basis

    My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...
  18. Bobman

    Determine a basis for kernel/nullspace

    Homework Statement [/B] A=##\begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \end{bmatrix}## Question: Determine a basis for ker(A).Homework Equations Ker(A)=##\begin{bmatrix} 1 & 2 & 3& |&0 \\ 4 & 8 & 12& |&0 \end{bmatrix}## (And reduce with gaussian elimination) The Attempt at a Solution (Sorry...
  19. Hiero

    I Is there any theoretical basis for laws being 2nd order

    Hi, I’m just wondering about this: Are there any theoretical reasons why physical laws take the form of 2nd order (in time) differential equations? Or is it just observed to be that way? Are there ANY laws (even in a limited context) which are 3rd (or higher) order in time?
  20. M

    MHB Exploring Basis Subsets in a 5-Element Vector Space

    Hey! :o Let $V$ be a vector space with with a 5-element basis $B=\{b_1, \ldots , b_5\}$ and let $v_1:=b_1+b_2$, $v_2:=b_2+b_4$ and $\displaystyle{v_3:=\sum_{i=1}^5(-1)^ib_i}$. I want to determine all subsets of $B\cup \{v_1, v_2, v_3\}$ that form a basis of $V$. Are the desired subsets the...
  21. M

    MHB Sum of basis elements form a basis

    Hey! :o Let $V$ be a vector space. Let $b_1, \ldots , b_n\in V$ and let $\displaystyle{b_k':=\sum_{i=1}^kb_i}$ for $k=1, \ldots , n$. I want to show that $\{b_1, \ldots , b_n\}$ is a basis of $V$ iff $\{b_1', \ldots , b_n'\}$ is a basis of $V$. I have done the following: Let $B:=\{b_1...
  22. X

    I Bloch Sphere Change of Basis

    Anyone know how to change a basis of a qubit state of bloch sphere given a general qubit state? There are 3 different basis corresponding to each direction x,y,z where |1> ,|0> is the z basis, |+>, |-> is the x basis and another 2 ket notation for y basis. Given a single state in the x basis...
  23. M

    I Completeness of a basis function

    Hi PF! I'm somewhat new to the concept of completeness, but from what I understand, a particular basis function is complete in a given space if it can create any other function in that space. Is this correct? I read that the set of polynomials is not complete (unsure of the space, since Taylor...
  24. O

    Change of basis computation gone wrong....

    Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...
  25. H

    I Represenation of a state vector in a different basis

    Is it possible to expand a state vector in a basis where the basis vectors are not eigenvectors for some observable A? Or must it always be the case that when we expand our state vector in some basis, it will always be with respect to some observable A?
  26. M

    Prove that the standard basis vectors span R^2

    Homework Statement I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question? Standard basis vectors: e_1, e_2 or i,j
  27. S

    B GramSchmidt process for Taylor basis

    Why are the limits as so for the integral?
  28. olgerm

    I How are basis vectors defined?

    hi. if I know how to convert coordinates from a system to cartesian system, then how can I find basevectors of that coordinatesystem? Is it possible that basevectors are different in different points(with different coordinates)? What is most general definition of basevectors? I tought it would...
  29. Jd_duarte

    I Orthonormal Basis of Wavefunctions in Hilbert Space

    Hello, I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...
  30. S

    I Historical basis for: measurement <-> linear operator?

    What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?
  31. karush

    MHB 307.8.1 Suppose Y_1 and Y_2 form a basis for a 2-dimensional vector space V

    nmh{796} $\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$ $\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$ $$Y_1=\begin{bmatrix}a\\b\end{bmatrix} \textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$ $\textit{ then }$...
  32. M

    MHB Show that there is a basis C of V so that C* = Λ

    Hey! :o Let $K$ be a field and $V$ a $n$-dimensional $K$-vector space with basis $B=\{b_1, \ldots , b_n\}$. $V^{\star}$ is the dual space of $V$. $B^{\star}$ is the dual basis corresponding to $B$ of $V^{\star}$. Let $C=\{c_1, \ldots , c_n\}$ be an other basis of $V$ and $C^{\star}$ its...
  33. evinda

    MHB Find components of map with respect to basis

    Hello! (Wave) Let linear map $f: \mathbb{R}^3 \to \mathbb{R}^2$, $B$ basis (unknown) of $\mathbb{R}^3$ and $c=[(1,2),(3,4)]$ basis of $\mathbb{R}^2$. We are given the information that $cf_s=\begin{pmatrix} 1 & 0 & 1\\ 2 & 1 & 0 \end{pmatrix}$. Let $v \in \mathbb{R}^3$, of which the coordinates...
  34. M

    MHB Basis of kernel and image

    Hey! :o Let $A\in \mathbb{C}^{2\times 2}$ and $L_A:\mathbb{C}^{2\times 2}\rightarrow \mathbb{C}^{2\times 2}, \ X\mapsto A\cdot X$. We consider the matrix \begin{equation*}A=\begin{pmatrix}-1 & 2 \\ 2 & -4\end{pmatrix}\end{equation*} and the basis \begin{equation*}B=\left \{\begin{pmatrix}1...
  35. T

    MHB Basis for set of solutions for linear equation

    [solved] Basis for set of solutions for linear equation Hi, I have this problem I was working through, but I'm not sure that I've approached it from the right way. The problem consists of 3 parts, which build off of each other. I'm pretty confident about the first two parts, but no so much...
  36. S

    Gram-Schmidt for 1, x, x^2 Must find orthonormal basis

    Homework Statement Find orthonormal basis for 1, x, x^2 from -1 to 1. Homework Equations Gram-Schmidt equations The Attempt at a Solution I did the problem. My attempt is attached. Can someone review and explain where I went wrong? It would be much appreciated.
  37. Pushoam

    I Expanding a given vector into another orthonormal basis

    Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis. In equation 9.2.32, the basis of ## |V \rangle## is not given. ## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ? Do...
  38. M

    Dot product and basis vectors in a Euclidean Space

    Homework Statement I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space. Homework EquationsThe Attempt at a Solution The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...
  39. M

    Mathematica Can I Scale Down Basis Functions Without Losing Zero Force?

    Hi PF! I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...
  40. S

    I Non-coordinate Basis: Explained

    Hello! I read about this in several place, but I haven't found a really satisfying answer, so here I am. As far as I understand, non-coordinate basis are mainly obtained from coordinate basis, by making the system orthonormal. For example the unit vector in polar coordinates in the direction of...
  41. P

    I Why choose traceless matrices as basis?

    While writing down the basis for SU(2), physicists often choose traceless hermitian matrices as such, often the Pauli matrices. Why is this? In particular why traceless, and why hermitian?
  42. A

    A Rigorous transition from discrete to continuous basis

    Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...
  43. StevieTNZ

    B Polarization Basis: Measuring Entangled Photons in 45/135

    I have a quick, general, query. When conducting an experiment with entangled photons, measuring in the +/- basis would mean in the 45/135 basis?
  44. M

    Derivative of basis vectors

    Homework Statement I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates Homework EquationsThe Attempt at a Solution differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.
  45. L

    B How to find the dual basis vector for the following

    ei=i+j+2vk , how to find the dual basis vector if the above is a natural base?
  46. bhobba

    A Physical Basis of Lovelock's Theorem: GR & Equivalence Principle

    This came up in another thread. GR more or less follows directly from Lovelock's Theorem. You simply assume the metric has a Lagrangian. Where does that leave other things like the Equivalence principle? Thanks Bill
  47. K

    I Basis Vectors & Inner Product: A No-Nonsense Introduction

    I read from this page https://properphysics.wordpress.com/2014/06/09/a-no-nonsense-introduction-to-special-relativity-part-6/ that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...
  48. R

    From a given basis, express a polynomial

    Homework Statement Express a polynomial in terms of the basis vectors. {x2 + x, x + 1, 2} Homework Equations 3. The Attempt at a Solution [/B] I think the answer is: (x2+x)^2 + (x + 1) + 2 = 0 simplified to become: x4 + 2x3 + x2 + x + 3 = 0
  49. K

    I Normalized basis when taking inner product

    Consider that a vector can be represented in two different basis. My question is do we need to normalize both basis before taking the inner product? What motivates this question is because I found out that the inner product of a vector having components ##a,b## in the normalized polar basis of...
  50. G

    MHB Problem about equivalent conditions for a basis of a free module

    Here's the problem: I don't see why there should be only finitely many nonzero a_z in b. I was able to prove uniqueness assuming that there only finitely many nonzero. I was able to show b implies d and b implies c, c implies a.
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