Basis Definition and 1000 Threads

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. T

    Change of Basis between different size spaces

    Hi, I'm working on an example question with the following info: \alpha = {(3,0,1) , (3,1,1), (2,1,1)} \beta = {(1,1), (1,-1)} Are a set of bases. [T]\beta\alpha = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say: Let T: R3--> R3 be the transformation whose...
  2. G

    Question about relating basis of curvilinear coordinate systems.

    Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi http://en.wikipedia.org/wiki/Curvilinear_coordinates Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian...
  3. D

    RElation of partial differential operator and Basis vector

    Hi everyone: How is the following derived? Just for example: \Deltax\alphae\alpha=\Deltax\alpha(\delta/\deltax\alpha) does it not mean? e\alpha=\delta/\deltax\alpha But How?
  4. F

    Calculate the discriminant of a basis [Number Theory]

    Question: The needed proposition and two examples: This is as far as I have got: I need to reduce this (I think) so I can represent is as a matrix! Any idea on how to do this? Thanks
  5. matqkks

    Change Basis: Reasons to Write Vector in Rn Other Than Standard Basis

    Why should we want to write a vector in Rn in other than standard basis? A normal application of linear transformations in most textbooks is converting a given vector in standard basis to another basis. This is sometimes a tedious task. Why carry out this task? Thanks for your replies in advance.
  6. G

    Finding Basis of Null Space and Range

    Homework Statement Prove T is a linear transformation and find bases for both N(T) and R(T). Homework Equations The Attempt at a Solution T:M2x3(F) \rightarrow M2x2(F) defined by: T(a11 a12 a13) (a21 a22 a23) (this is one matrix) = (2a11-a12 a13+2a12)...
  7. F

    The General Linear Group as a basis for all nxn matrices

    I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F). I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional...
  8. S

    Expectation values of QHO in |n> basis

    Is it possible to express ANY observable A(X,P) in terms of the ladder operators? I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be...
  9. T

    Dimension of Vector Space R^X: Finite/Infinite Cases

    The finite case is fine, as a vector space it is easy to show that R^X is isomorphic to R^n. What about when X is infinite? I believe it is true in general that dim(R^X) = #(X), which I hope holds in the infinite case too. I know that the set given by B={b_x; x in X} defined as b_x(y) =...
  10. M

    Density of State for a two atom basis

    Homework Statement I have a problem on my assignment in which I am required to find the specific heat of a two atom basis (diatomic) using the Debye model. My problem is coming up with the density of states for a diatomic setup in 1D. Homework Equations Density of state...
  11. C

    Video Games and Their Basis in Theory

    Are there any theories with which the mass effect fields of Mass Effect and the slipstream space of Halo draw inspiration from? Is there any scientific basis for either, or is it complete fantasy?
  12. K

    Question regarding basis of function space

    I only possesses a rudimentary understanding of Linear Algebra so I'm not going to be rigorous in my explanation, but is the concept of an infinite basis well defined? More specifically, I was thinking about how the polynomials could form a basis for function space, given that every function has...
  13. A

    Sobolev type norms and basis functions

    Hello everybody, I am given a "Sobolev type innerproduct" \langle f,g \rangle_{\alpha} = \langle f,g \rangle_{L^2} + \alpha \langle Rf,Rg \rangle_{L^2} for some \alpha \geq 0 and R some differential operator (e.g. the second-derivative operator). My question is now whether a function...
  14. S

    How to Find the B Matrix in Matlab for Wavelet Basis Functions?

    Hi all, Say that I have a 1D signal such that f=Bw where f is the signal B is the basis functions and w is the wave co-efficients. The question that I have is how do I find the B matrix in Matlab. I am looking through WaveLab and Rice Wavelet packages but simply cannot find an answer. As...
  15. P

    Do Columns of M² Form a Basis?

    Hi everyone, This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem. Let β=\{ v_1,v_2,...,v_n\} be a basis for R^n . Let M be the matrix whose columns are the basis vectors in β. Do the...
  16. S

    What is the basis for a vector space in Structural Engineering?

    I'm currently doing a self study course on Linear Algebra. Can anyone give me an example of vector space and basis with reference to Structural Engineering? For example I have a displacement vector for a simply supported beam as: [thata_a theta_b]^T where; theta_a and theta_b...
  17. S

    What is the basis for the theory that WIMPs could be detected by

    What is the basis for the theory that WIMPs could be detected by seeing a vibration in the atomic nucleus of normal matter? If they (all) really do interact so weakly, why do scientist think they might be able to detect just a few?? an explanation in layman's terms would be great. thanks?
  18. S

    Finding basis functions for approximating transcendental function

    I am working on a problem where I want to approximate a transcendental function of the form f(x) = x^Ne^{x} for x \geq 0 as a linear combination of functions of the form x^v \text{where} -1 < v < 0. How can I find the basis functions of the desired form to represent my transcendental...
  19. B

    What is the Basis of a Quotient Ring?

    In my Abstract Algebra course, it was said that if E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}. The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}]. But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n...
  20. A

    Tight binding method for a 1D crystal with a diatomic basis

    Hi, I'd be most grateful for any help regarding the following problem: Consider a 1D crystal with 2 atoms in a primitive cell (let's call them atoms A and B). Each atom has only one valence orbital denoted as \left|\psi_A(n)\right> and \left|\psi_B(n)\right> respectively. Show that the...
  21. A

    Exploring Change of Basis Matrices and Intuitive Examples in Linear Algebra

    Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates...
  22. L

    Change of basis matrix(linear algebra)

    Hi I'm stuck on this problem and I could not find similar examples anywhere.. any help would be greatly appreciated, thank you. Homework Statement Compute the change of basis matrix that takes the basis V1 = \begin{bmatrix} -1 \\ 3 \end{bmatrix} V2 = \begin{bmatrix} 2 \\ 5 \end{bmatrix}...
  23. C

    Prove coordinate vectors are unique for given basis

    Homework Statement Prove that the coordinates of a vector v in a vector space Vn are unique with respect to a given basis B={b1,b2,...,bn} Homework Equations The Attempt at a Solution not sure at all what to do with this
  24. C

    Problem in constructing Matrix representation in |↑↓> basis

    If I want to derive the matrix representation for operator Q in the |S1=1/2 ,m1> |S2=1/2 ,m2 > basis, where |Si,mi> are common eigenstates of S2 , Si,z for the ith particle. And I do it in this way: <↑↑|Q|↑↑> <↑↑|Q|↑↓> <↑↓|Q|↓↑> <↑↑|Q|↓↓> <↑↓|Q|↑↑> <↑↓|Q|↑↓> <↑↓|Q|↓↑> <↑↓|Q|↓↓> ...
  25. B

    Linear Algebra: does it form a basis?

    Homework Statement See attachment. The Attempt at a Solution I already did parts i and ii (correctly, I hope). On part iii I found 2 linearly independent elements to be: t+1, t^2 - 1. However, I don't understand how to show that these form a basis of W. Because W is a subspace of P2, and P2...
  26. E

    Understanding Basis for Solving Linear Algebra Problems

    Hey guys There are so many of these damn "Find a basis" questions and I can't get any of them because we never directly learned how...or she never showed us in class...my final exam is tomorrow. Here are some examples of questions: http://184.154.165.18/~devilthe/uploads/1323453294.png...
  27. V

    Finding orthonormal basis for the intersection of the subspaces

    Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach
  28. A

    Is {u+v+w, v+w, w} Also a Basis for V?

    Let u,v,w\in V a vector space over a field F such that u≠v≠w. If { u , v , w } is a basis for V. Prove that { u+v+w , v+w , w } is also a basis for V. Proof Let u,v,w\in V a vector space over a field F such that u≠v≠w. Let { u , v , w } be a basis for V. Because { u , v , w } its a basis...
  29. O

    Values for which a set of vectors form a basis of Rn

    Homework Statement For what value(s) of λ is the set of vectors {(λ^2-5, 1, 0), (2, -2, 3), (2, -3, -3)} form a basis of ℝ^3Homework Equations in order for a vector to form a basis it has to span R3 and the set has to be linearly independent.The Attempt at a Solution i tried doing row...
  30. F

    Is the Set of Functions f[sub k] a Basis for the Vector Space V?

    Homework Statement Let S be any non-empty set, F be a field and V={ f : S -> F such that f(x) = 0 } be a vector space over F. Let f[sub k] (x) : S -> F such that f[sub k] (x) = 1 for k=x, otherwise f[sub k] (x) = 0. Prove that the set { f [sub k] } with k from S is a basis for the vector space...
  31. L

    Change for position to energy basis

    Homework Statement Give expressions for computing the matrix elements Xmn of the matrix X representing the position operator X in the energy basis (using eigenvectors of the Harmiltonian operator) Also told to consider the example of the harmonic oscillator where energy eigenvalues are...
  32. H

    [Linear Algebra] Basis, Linear Inependence

    Homework Statement Homework Equations The Attempt at a Solution 2) No clue.
  33. M

    Relationship of Basis to Dual Basis

    If we're working in R^n and we consider the elements of a basis for R^n to be the column vectors of an nxn invertible matrix B, then what is the relationship between B and the matrix whose row vectors represent elements of the corresponding dual basis for R^n*? My guess, which Wikipedia helped...
  34. F

    Find Basis for Span: \vec{a_{1}} to \vec{a_{4}}

    I don't wan't a solution I wan't only instructions how to solve this problem: Find a basis for the span: \vec{a_{1}}=(1,\,-1,\,6,\,0),\,\vec{a_{2}}=(3,\,-2,\,1,\,4),\,\vec{a_{3}}=(1,\,-2,\,1,\,-2),\,\vec{a_{4}}=(10,\,1,\,7,\,3)
  35. Z

    Schools What to do about grad school? What does an engineer do on a day to day basis?

    I’m currently a physics/math major. I work very hard and am proud of my 4.0 GPA. However, as my peers and professors begin to talk about grad school I realize I don’t have a clue what I'm supposed to do. My goal is to go to Penn State for an advanced degree in some type of engineering or...
  36. L

    Inner product space and orthonormal basis.

    Homework Statement Assume the inner product is the standard inner product over the complexes. Let W= Spanhttp://img151.imageshack.us/img151/6804/screenshot20111122at332.png Find an orthonormal basis for each of W and Wperp.. The Attempt at a Solution Obviously I need to use Gram-Schmidt...
  37. V

    A problem on finding orthogonal basis and projection

    Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...
  38. V

    A problem on finding orthogonal basis and projection

    Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...
  39. V

    Is the ISW Hamiltonian Diagonal in the Energy Basis?

    Homework Statement Find the matrix elements of the Hamiltonian in the energy basis for the ISW. Is it diagonal? Do you expect it to be diagonal? Homework Equations H=\frac{p^2}{2m}+V \frac{d}{dt}\langle Q \rangle = \frac{i}{\hbar} \langle[\hat H, \hat Q] \rangle + \langle...
  40. A

    Finding the basis of a null space

    Homework Statement The matrix is: -2 -2 -4 4 -1 1 2 -2 -1 0 -3 0 -4 1 -7 -2 I know the dimensions for the null space are 2 Homework Equations I know that to find the basis for a null space Ax=0, so I row reduced it and I got 1 0 3 0 0 1 5 -2 0 0 0 0 0 0 0 0 The Attempt...
  41. M

    What is the basis for a plane perpendicular to 3x + 2y − z = 0?

    Homework Statement Let W be the plane 3x + 2y − z = 0 in ℝ3. Find a basis for W perpendicularHomework Equations The Attempt at a Solution I thought a basis for this plane could be generated just by letting x=0 and y=1, finding z and then doing the same thing but this time letting x=1 and y=0...
  42. G

    Mathematical Basis for Learning Relativity

    Hi, I seem to remember there is a book by Steven Weinberg that gives the mathematical basis for tensor calculus for relativity, but the name escapes me. Anyone know what I'm talking about?
  43. S

    Are super-operators always writable in a basis independent form?

    in particular, i wonder if the trasposition super operator is basis independent or not. We can always write an operator W as \hat{W}=\sum_{i,j} c_{i,j} |i\rangle\langle j| and for the transposed we obtain \hat{W}^T=\sum_{i,j} c_{j,i} |i\rangle\langle j| we obtain a relation true for each...
  44. D

    Basis for the orthogonal complement.

    Homework Statement Let W be the plane 3x + 2y - z = 0 in R3. Find a basis for W^{\perp}Homework Equations N/A The Attempt at a Solution Firstly, I take some arbitrary vector u = \begin{bmatrix}a\\b\\c\end{bmatrix} that is in W^{\perp}. Then I note that W can be rewritten in terms of the...
  45. Matterwave

    Pondering basis vectors and one forms

    So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey...
  46. S

    Find Basis for Subspace of R^4 Spanned by Vectors

    Hello. First, I'd like to apologize because I don't know where to go ask for homework on linear algebra on the forums so if anyone could please let me know, that would be appreciated. Here's the question: Find a basis for the subspace of R^4 spanned by the given vectors Here's the answer...
  47. N

    What is the position operator in the momentum basis for a given momentum value?

    Homework Statement I need to prove that, <p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'} i.e. find the position operator in the momentum basis p for p'... It's easy to prove that <x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'} (position operator in position basis for x') since I...
  48. F

    Partial derivative of radial basis function

    Homework Statement Calculate the partial derivatives (∂f/∂x & ∂f/∂y) Homework EquationsThe Attempt at a Solution really confusing me with the use of the summation and power to 3/2. This is my attempt, most definitely wrong but still tried. ∂f/∂x = x + c1*(2*(x-x1))*([( x-x1 )^2 +...
  49. W

    How to Express the Maxwell Field Using Wedge Products?

    Homework Statement The problem is to write it in terms on coordinate basis using the wedge product, F=F_{\mu\nu}dx^{\mu}\wedge dx^{\nu} from the basis with the tensor prdouct. Homework Equations The EM field strength tensor can be written, with a coordinate basis, F=F_{\mu\nu}dx^{\mu}\otimes...
  50. P

    Unveiling the Physical Basis of Fock Space Dynamics for Chemists

    Hello, I am a chemist and have been working on chemical dynamics. Recently I have started working on some many body interactions. Therein I have found some ideas about Fock Space, Fock Matrix, Fock Space Coherences. These are extensively used to provide characteristic information in...
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