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. H

    Finding the basis of a subspace

    Homework Statement How do I find a basis for: the subspace of R^3 consisting of all vectors x such that x ⋅ (1,2,3) = 0. Homework Equations I believe this is performed through setting x = x,y,z, setting each parameter sequentially equal to 1 while the others are set to o, putting into a matrix...
  2. M

    Extending a Set of 3 Orthogonal Vectors to a Basis in R^5

    I've attached the question to this post. The answer is true, but I'm trying to figure out why. Using Gram-Schmidt, I can only necessarily find 3 orthogonal vectors given 3 linearly independent vectors from ## R^5 ##. How then is it possible to extend this set of 3 vectors that are linearly...
  3. M

    Classifications of basis vectors

    Hello every one , in this pic i just printed ( Tensors_The Mathematics of Relativity Theory and Continuum Mechanics by Anadijiban Das ) here the author classifies the basis into 3 types 1- is the general basis (non-holomonic ) , 2- coordinate basis ( holomonic ),3- orthonormal basis ( non-...
  4. emeriska

    Convert Hamiltonian to matrix in weird basis

    Hi guys, I'm having a hard time with that one from Cohen-Tannoudji, ##F_{VI}## # 6. I'm translating from french so sorry if some sentence are weird or doesn't use the right words. 1. Homework Statement We consider a system of angular momentum l = 1; A basis from it sub-space of states is...
  5. PsychonautQQ

    Question about row space basis and Column space basis

    Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)> By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S. Cool, not...
  6. T

    Change of Basis With Orbital Angular Momentum

    Homework Statement We have the initial orbital angular momentum state in the x basis as |l,ml>x=|1,1>x. We are asked to find the column vector in the z-basis that represents the initial orbital angular momentum of the above state. It then says "hint: use an eigenvalue equation". Homework...
  7. Math Amateur

    MHB Free Modules w/Multiple Bases: 2nd Issue w/Example 5 (Paul E. Bland)

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with Example 5 showing a module with two bases ... ... Thanks to Caffeinemachine, I have largely clarified one issue/problem I had with Example 5, but now have a...
  8. Math Amateur

    MHB Free Modules With More Than One Basis - Bland - Example 5, page 56

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with Example 5 showing a module with two bases ... ... Example 5 reads as follows:I am having trouble understanding the notation and meaning of M = \bigoplus_{...
  9. A

    Finding a basis for a particular subspace with Dot Product restrictions

    Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then, w-2x+3z=0 and 2x+y+3z=0 So, she solves the system and get the following: Subspace= {...
  10. W

    Full Basis of Angular Momentum Eigenstates

    Homework Statement Consider the case of an atom with two unpaired electrons, both of which are in s-orbitals. Write the full basis of angular momentum eigenstates representing the coupled and uncoupled representations Homework Equations l=r×p lx=ypz-zpy ly=zpx-xpz lz=xpy-ypx l+=lx+ily...
  11. Logan Land

    MHB Finding the Rank and Basis for a matrix

    Find rank and the bases for column and row spaces of the matrices 1 0 1 2 -1 3 3 -1 4 Now I can see instantly that row 3 is just row 1 + row 2 so it must be dependent. So that means that row 3 will turn to a row of zeros and thus the rank(A)=2 if I reduced matrix A to row echelon it...
  12. Logan Land

    MHB Find basis and dimension of the intersection of S and T

    If you have a vector space S be spanned by vectors (x1,y1,z1), (x2, y2, z2), (x3, y3, z3) and T spanned by (x1,y1,z1), (x2, y2, z2), (x3, y3, z3). How would you find the basis and dimension of the intersection of S and T . (x,y,z can be any value) Do I go about it like this? a(x1,y1,z1)+b(x2...
  13. S

    How do I make a change of basis with tensors in multilinear algebra?

    I did some linear algebra studies and learned how to change between foreign bases and the standard basis: Change of basis matrix multiplied by the vector in coordinates with respect to the foreign basis equals the vector in coordinates with respect to the standard basis. Of course, this is...
  14. Logan Land

    MHB Dimension and Basis of a Relation in R5

    Find dimension and basis of the set of all points in R5 whose coordinates satisfy the relation x1 +x2 +x3 +x4 =0. shouldn't there be 5 vectors to satisfy the basis since they are asking about R5? but the relation only has x1, x2, x3, and x4. or would my matrix just look like this 1 0 0 0 0 0...
  15. Logan Land

    MHB Find coordinates of f′, f′′, f′′′ in the basis

    Consider the polynomial f(x) = x^5 − 5x^4. (a) Find coordinates of f′, f′′, f′′′ in the basis {1, x, x2, x3, x4, x5} I no f ' = 5x^4-20x^3 f " = 20x^3-60x^2 and f "' = 60x^2-120x but from there where to begin? do I make a matrix of like the following? 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0...
  16. perplexabot

    How Do You Change the Basis of an Operator in Linear Algebra?

    Homework Statement Homework Equations \check{T} = BTB^{-1} (eq1) The Attempt at a Solution Ok, so I have a couple of questions here if I may ask... First, I want to be sure I understand the wording of (a) and (b) correctly. Is the following true?: (a) Write the matrix T...
  17. S

    How do you derive relativistic tensors in an orthonormal basis?

    I have been recently trying to derive the Einstein tensor and stress energy momentum tensor for a certain traversable wormhole metric. In my multiple attempts at doing so, I used a coordinate basis. My calculations were correct, but the units of some of the elements of the stress energy momentum...
  18. T

    Writing Hamiltonian in the basis

    Homework Statement H = \frac{2e^2}{\hbar^2 C} \hat{p^2} - \frac{\hbar}{2e} I_c cos\hat\theta , where [\hat\theta , \hat{p}] = i \hbar How can we write the expression for the Hamiltonian in the basis |\theta> Homework EquationsThe Attempt at a Solution I have already solved most part of...
  19. L

    How Is the CsCl Unit Cell Described Using Its Primitive Cell and Basis?

    Homework Statement CsCl has a BCC unit cell containing 2 atoms, Cs in the origin of the cell and Cl at the centre of the cell. Describe the CsCl unit cell in terms of its primitive cell+basis. Homework Equations R = n1a1 + n2a2 + n3a3 ~ R is the vector which relates one point of the lattice...
  20. gfd43tg

    Concentration of species based on what basis is used

    Homework Statement For the gas phase reaction with an equimolar feed of ##N_{2}## and ##H_{2}## ## \frac {1}{2} N_{2} + \frac {3}{2} H_{2} \rightarrow NH_{3}## If you took ##N_{2}## as your basis of calculation, could 60% conversion of ##N_{2}## be achieved? Homework EquationsThe Attempt at...
  21. B

    Change of Basis For Pauli Matrix From Z Diagonal to X Diagonal Basis

    I want to find a matrix such that it takes a spin z ket in the z basis, | \; S_z + >_z and operates on it, giving me a spin z ket in the x basis, U \; | \; S_z + >_z = | \; S_z + >_x I would have thought that I could find this transformation operator matrix simply by using the...
  22. arpon

    How to define covariant basis in curved space 'intrinsicly'?

    In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e. ##∂R/(∂z^i )=z_i## But in curved space (such as, the two dimensional space on a sphere) how can we define covariant basis 'intrinsicly'?(as we have no position...
  23. C

    Finding an orthonormal basis for a subspace

    Homework Statement Find an orthonormal basis for the subspace of V4 spanned by the given vectors. x1 = (1, 1, 0, 1) x2 = (1, 0, 2, 1) x3 = (1, 2, -2, 1) Homework Equations Gram-Schmidt Process The Attempt at a Solution I have used the Gram-Schmidt process but seem to be running into trouble...
  24. W

    Completeness of basis in quantum mechanics

    In a QM course, I learn that an operator can be represented by basis vectors If the basis vector is complete, the following relation holds There exist coefficient Mij such that Sigma Mij |i > < j|. = I , |i> is the basis! and I is the identity matrix But isn't that in linear algebra We...
  25. L

    Info on Bessel functions & their use as basis functions.

    Hello all,As an exercise my research mentor assigned me to solve the following set of equations for the constants a, b, and c at the bottom. The function f(r) should be a basis function for a cylindrical geometry with boundary conditions such that the value of J is 0 at the ends of the cylinder...
  26. S

    Compound Interest Paid in 3.5 Years: Regional Laws & Concepts

    - This may be a stupid question as I am totally new to the concept of interest. I don't even know if my question is valid. - Figure is given below for referenceSuppose I deposit some money in a bank that pays compound interest on yearly basis. If I decide to withdraw my amount at the end of 3.5...
  27. R

    Is the Canonical Basis in Mathematics Just a Useful Convention?

    In many places the canonical basis is defined as a set of vectors with coordinates as: \boldsymbol{e}_i=(0,...,1,...0) where "1" is on the i-th place. In my undestanding of such definicion every basis is canonical basis. If we write coordinates of basis vectors in the same basis we will get such...
  28. A

    1000 Basis functions limitation - Gaussian 09?

    Does anyone know if there is a limitation of the number of basis functions for freq calculation in Gaussian 09? I have 2 freq calculations, one has 879 basis functions and the other one has 1047 basis functions. Gaussian 09 couldn't finish the 1047 basis function calculation no matter how...
  29. I

    Linear Algebra orthogonal basis and orthogonal projection

    I was placed into honors calculus III for school. I was happy about this and I consider myself to be a pretty quick learner in math. However, my teacher is using many notations and terms that I am completely unfamiliar with. Mostly, I believe, because I've never taken linear algebra. I am...
  30. A

    Curvilinear basis in spherical polar coordinates

    Homework Statement As a part of my self study I am trying to find the covariant basis vectors in the spherical polar coordinates. Since I have never done anything like this before I would appreciate if someone could tell me whether I am on the rigth track. Homework Equations...
  31. D

    Linear operators and change of basis

    Following on from a previous post of mine about linear operators, I'm trying to firm up my understanding of changing between bases for a given vector space. For a given vector space V over some scalar field \mathbb{F}, and two basis sets \mathcal{B} = \lbrace\mathbf{e}_{i}\rbrace_{i=1,\ldots ...
  32. D

    Orthonormal Basis: Show A is Self-Adjoint

    Hi. Been looking at a question and its solution and I'm confused. Question is - Let ##ψ_n## ,n=1,2,... be an orthonormal basis consisting of eigenstates of a Hamiltonian operator H with non-degenerate eigenvalues ##E_n##. Let A be a linear operator which acts on the energy eigenstates ##ψ_n##...
  33. fluidistic

    Density matrix, change of basis, I don't understand the basics

    Homework Statement Hello people, I am trying to understand a problem statement as well as the density operator, but I still don't get it, desperation is making me posting here. The problem comes as The problem then continues with other questions but I'm having troubles with the very first one...
  34. E

    MHB Notation for vector coordinates in a given basis

    Sorry for a long post. I am looking for a clear and concise way to explain how to compute coordinates when changes of basis or linear operators are involved. I would like to avoid the summation notation as much as possible and use the definition of matrix multiplication only in the beginning...
  35. Math Amateur

    MHB Similar Matrices and Change of Basis

    I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence. On page 419 D&F define similar matrices as follows: They then state the following: BUT? ... how...
  36. M

    A Bit Confused About Polar Basis Vectors

    Let me say from the beginning I'm not talking about the non-coordinate unit vectors for polar coordinates. I'm talking about basis vectors. Let me just ask it as boldly as possible: how does one use these basis vectors in order to describe a vector? I know they are different at every point, so...
  37. D

    Bases of vector spaces and change of basis

    Hi all, Just doing a bit of personal study on vector spaces and wanted to clear up my understanding on the following. This is my description of what I'm trying to understand, is it along the right lines? (apologies in advance, I am a physicist, not a pure mathematician, so there are most...
  38. K

    MHB Find matrix representation with respect to the basis

    The set of all solutions of the differential equation \d{^2{y}}{{x}^2}+y=0 is a real vector space V=\left\{f:R\to R \mid f^{\prime\prime}+f=0\right\} show that \left\{{e}_{1},{e}_{2}\right\} is a basis for $V$, where {e}_{1}:R \to R, \space x \to \sin(x) {e}_{2}:R \to R, \space x \to...
  39. K

    MHB Using the given basis of a vector to prove other basis

    Hi Guys having a bit of trouble understanding vector basis. If \left\{{e}_{1},{e}_{2},{e}_{3}\right\} is a basis for vector space $V$ over the field $F$ and {f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3} how can I go about proving that...
  40. S

    How Can I Write an Operator in the Coupled and Uncoupled Basis?

    Given a system of two identical particles (let's say electrons), of (max) spin 1/2 (which means the magnetic quantum number of each of the electrons can be either 1/2 or -1/2), how can we write the operators (total angular momentum, z-component of the total angular momentum etc.) (a) in the...
  41. P

    Changing the basis of Pauli spin matrices

    Homework Statement Find the matrix representation of S_z in the S_x basis for spin 1/2. Homework Equations I have the Pauli matrices, and I also have the respective kets derived in each basis. There aren't really any relevant equations, other than the eigenvalue equations for the...
  42. B

    Can non-compact 4-dimensional spacetimes have a global tetrad?

    Suppose you have a differentiable manifold where at each point you have attached a set of basis vectors X_1,X_2,...,X_n. One thing that I don't have clear is the difference between a coordinate basis and a non-coordiante basis. I've been told that there is a way to check if the set of basis...
  43. carllacan

    Matrix elements of position operator in infinite well basis

    Homework Statement Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.Homework Equations The Attempt at a Solution Tell me if I'm right so far (the |E> are the eigenkets) X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle...
  44. C

    Variation of the Einstein-Hilbert action in noncoordinate basis

    The variation of the Einstein Hilbert action is usually done in coordinate basis where there is a crucial divergence term one can neglect which arise in the variation of the Ricci tensor, and is given by ##g^{ab}\delta R_{ab} = \nabla_c w^c## where $$w^c = g^{ab}(g^{db} \delta \Gamma^{c}_{db} -...
  45. L

    What is the basis of Rotation Curve?

    I have been reading about rotation curves, and the I understand the basics, but I am trying to understand the basis / meaning of the curves? Are the findings based on redshifts of individual stars / "bins of light" at set distances from the centre of the galaxy? And given that the...
  46. R

    How can we represent an image using basis images?

    I have read that using Fourier transformation we can decompose any arbitrary image into othogonal basis images and reconstruct it back. But i don't understand terms like "othogonal " and "basis image". So can anybody shower their ideas on the above terms with example ??
  47. P

    Resolving Vector in Arbitrary Basis: Proving Coefficients' Independence

    Hi, I came across this statement while I was reading an article on resolution of vector in arbitrary basis. "v = αa + βb + γc ---(1) , where a, b and c are three independent vectors. we observe that the coefficient α cannot involve any overlap of v with either b or c ; β cannot involve any...
  48. M

    Dual basis and kernel intersection

    The problem statement, all variable Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that ##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##. The attempt at a solution. For ##→##: Let ##\{v_1,...,v_n\}## be...
  49. Sai-

    Basis for a Plane that is given.

    Homework Statement Consider the plane 3x1-x2+2x3 = 0 in R3. Find a basis for this plane. Hint: It's not hard to find vectors in this plane. Homework Equations Plane: 3x1-x2+2x3 = 0 in R3. The Attempt at a Solution Let, A = \left[3,\right.\left.-1,\right.\left.2\right] \rightarrow...
  50. N

    Basis for Subspace: Find & Check LI

    Homework Statement Find the basis for the subspace 4x+y-3z The Attempt at a Solution I found that the basis is {[1;-4;0],[0;3;1]}. How do I know if it is linearly independent? I know that the mathematical definition of what LI is but how can it be applied to show in this case?
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