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. B

    How to Extend and Calculate a Basis for the Whole Space

    For example, say you start out with (2,1,0) and (2,0,2). Well the easiest answer here is to think of these two vectors in a plane, so you should take the cross product to get the vector that is not in the plane, and there you have a basis for R^3. But how about when we run into similar problems...
  2. L

    Quantum Mechanics. Normalised basis wave functions and Eigenvalues.

    Homework Statement Consider a particle with periodic boundary conditions of length L. Write dwon the expression for the normalised basis wave functions and their eigenvalues. Find the eigen value of the momentum and the expectation value of the momentum with respect to one of the momentum...
  3. H

    Proving (x_k+1, . . . , x_n) forms a basis for V/kerT

    Hi, I was working through this proof in my linear al textbook and there's this one step I can't get past. Any help would be appreciated. Homework Statement Let V be a finite dimensional vector space, and let T be a linear map defined on V. ker T \subseteq V and I am T \cong V/kerT Let...
  4. D

    Dimension of the image of a linear transformation dependent on basis?

    First of all I would like to wish a happy new year to all of you, who have helped us understand college math and physics. I really appreciate it. Homework Statement Determine the dimension of the image of a linear transformations f^{\circ n}, where n\in\mathbb{N} and...
  5. radou

    Discrete T1 space vs. locally finite basis

    Homework Statement The formulation of the problem confused me a little, so just to check. No T1 space has a locally finite space unless it is discrete. The Attempt at a Solution This means that, if X is a discrete T1 space, it has a locally finite basis, right? Btw, for the...
  6. B

    Dual basis and differential forms

    I was reading about dual spaces and dual bases in the book Linear Algebra by Friedberg, Spence and Insel (FSI) and they give an example of a linear functional, f_i (x) = a_i where [x]_β = [a_1 a_2 ... a_n] denotes the matrix representation of x in terms of the basis β = {x_1, x_2, ..., x_n} of...
  7. radou

    A countable basis vs. countably locally finite problem

    A "countable basis" vs. "countably locally finite" problem Homework Statement Sometimes it's fairly difficult to name a thread for a specific problem. :smile: So, one needs to show that, if X has a countable basis, a collection A of subsets of X is countably locally finite of and only if...
  8. F

    Representing a vector in terms of eigenkets in a continuos basis

    Representing a vector in terms of eigenkets in continuos basis(stuck here,guys help) I was reading Dirac and there is this formula which bothers me, |P>= \int{|\right{\xi'd}\rangle{d\xi'}} + \sum{|\right{\xi^{r}b}\rangle} Where |\right\xi'\rangle denotes the eigenket corresponding to the...
  9. M

    Proof of Basis Rep Theorem: Let n=a_0k^s+...+a_tk^t

    Hello, 1. Proof: Let denote the number of representations of to the base . We must show that always equals 1. (this means that we are trying to prove that there is only one representation?) Line 2. Suppose that n = a_{0}k^{s} + a_{1}k^{s-1} + ... + a_{s-t}k^{t} then...
  10. N

    Relationship between determinants and basis formation

    Hi, I'm scratching my head over the statement from my textbook which states when determinant is non-zero, the set of vectors blah blah is a basis for r^3. That does not make any sense to me because I know when a row of zeros in a matrix occur; the determinant is zero (through Gaussian...
  11. P

    Testing Change of Basis in Linear Operator

    I just want to test/verify my knowledge of change of basis in a linear operator.. (it's not a homework question). Suppose I have linear operator mapping R^2 into R^2, and expressed in the canonical basis (1,0), (0,1). Suppose (for the sake of discussion) that the linear operator is given by...
  12. C

    Find matrix of T with respect to the standard basis of R^3

    Homework Statement For this whole question let T be a linear transformation from R^3 to R^3 with T(1,0,0) = (2,2,2), T(0,1,0) = (1,2,2), T(0,0,1) = (0,0,1). (a) Find the image of (1,1,2009) (b) Find the matrix of T with respect to the standard basis in R^3 Homework Equations Standard...
  13. T

    Finding a Basis for the Kernel Space of a Matrix - Solving the RREF Method

    Homework Statement Find a basis for the kernel space of the following matrix: -1 -2 -1 2 2 -2 -4 -4 10 2 1 2 2 -5 2 -1 -2 0 -1 0 row reduce to 1 2 0 1 0 0 0 1 -3 0 0 0 0 0 1 0 0 0 0 0 Somehow read the solution as { [-2 1 0 0 0]T, [-1 0 3 1 0]T } .. I don't...
  14. T

    Simple basis / spanning set question (T / F)

    Homework Statement Let T: R4 --> R7 be a linear map whose kernel has the basis of v = [1 0 1 2]T. What is the dimension of the image of T? The Attempt at a Solution I have a very loose understanding of kernel and image, and am trying very hard to get this question. From my understanding, the...
  15. S

    Linear Algebra Question regarding basis of a kernel

    Homework Statement Let V be the vector space of all 2x2 matrices over Q V= {[x1 x2] : xi \in Q} ... x3 x4 Let A = [ -1 0 ] and let C:V --> V be the linear map C(X) = XA + AX .... -1 1 Find a basis for Ker(C) and a basis for Im(C) The Attempt at a Solution I used C(X) =...
  16. I

    How do I find the orthonormal basis for the intersection of subspaces U and V?

    Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg Solution in the image is right but my answer is so off from the current one. Homework Equations The...
  17. R

    Column Space Basis: Why Does Row Reduction Work?

    I am a bit puzzled by the following. You know how they teach you that in order to find column space you just need to row reduce the matrix, look at the columns with leading 1's in them and then just read off those columns from the original matrix? Well, why does that actually work? I'm trying to...
  18. S

    Jordan basis and jordan normal form

    Let A=2 -1 2 1 1 0 3 1 -2 1 0 1 -1 0 0 3 the characteristic polynomial of A is (x-1)3(x-2) find the minimal polynomial, jordan basis, and jordan normal form I know the minimum polynomial is (x-1)(x-2), but I am not sure how to find the nordan basis and jordan normal form
  19. M

    Finding a basis for the span of 4 vectors.

    Hi! Homework Statement I can't for the life of me figure out how to do this. I need to find the basis for the span of these four vectors: V1= 3, 1, -2, -4 V2 = -5, -3, 5, 9 V3 = 5, -1, 0, -2 V4 = -1, 5 -6 -8 2. The attempt at a solution I've figured out that the determinant is...
  20. murielg

    Find a BASIS for L(S) -The Image of S

    Homework Statement [PLAIN]http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs574.ash2/149609_293915114994_507054994_1176494_3477051_n.jpg Homework Equations The Attempt at a Solution Ok so I know that this plane goes thru the origim I guess to find the two column vectors that span...
  21. Saladsamurai

    Eigenvalues, Eigenspaces, and Basis

    Homework Statement Find the eigen values, eigenspaces of the following matrix and also determine a basis for each eigen space for A = [1, 2; 3, 4]Homework Equations \det(\mathbf{A} - \lambda\mathbf{I}) = 0 The Attempt at a Solution OK, so I found the eigenvalues and eigenspaces just fine...
  22. A

    Regarding normalization of the eigen basis vectors

    For a continuous eigen-basis the basis vectors are not normalizable to unity length. They can be normalized only upto a delta function. At the same time for discrete eigen basis the basis vectors are normalizable to unity length. What about the systems with both discrete as well as continuous...
  23. X

    Finding a Basis for Subspace in R^4: Linear Algebra Tips

    Help! Find a basis. Find a basis for the subspace of R^4 spanned by, S={(6,-3,6,340, (3,-2,3,19), (8,3,-9,6), (-2,0,6,-5) Figured I would set up the linear combination to test for independence.
  24. X

    Find a Basis for R^4 Subspace Spanning S

    Find a basis for the subspace of R^4 spanned by, S={(6,-3,6,340, (3,-2,3,19), (8,3,-9,6), (-2,0,6,-5) Not too sure where to start.
  25. B

    How to Find a Basis for W in P4[x] Using the Gram-Schmidt Process?

    I need some direction with respect to this problem please: Define the inner product on P4[x] over \Re as follows <f,g> = \int_{0}^{1}\f(x)g(x) dx let W be the subspace of P4[x] consisting of the poly. ) and all polynomials with degree 0, that is W =R Find a basis for...
  26. I

    Find Basis of R^n for Diagonal B Matrix of Reflection T in R^3

    Homework Statement Find a basis B of R^n such that the B matrix B of the given linear transformation T is diagonal. Reflection T about the plane x_1 - 2x_2 + 2x_3 = 0 in R^3. The Attempt at a Solution I just don't even know where to begin. I don't know how to interpret problem or how to...
  27. I

    What is the Basis and Dimension of a Polynomial Space with a Specific Condition?

    Got another linear space question. I'm getting closer to understanding what's going on, but I'm not there yet. Homework Statement Find a basis for the space and determine its dimension. The space of all polynomials f(t) in P2 such that f(1) = 0. Homework Equations The Attempt at...
  28. R

    Find a basis for the null space

    Homework Statement You're given two matrices (A and B). You want to find a basis for the space {x|x = Ay where By =0}. Homework Equations The Attempt at a Solution You're looking for all vectors x=Ay such that y is in the null space of B. So you're looking for a basis for only a part of...
  29. R

    Finding the basis for a subspace in vectorspace

    Homework Statement Find the basis for the subspace S of the vector space V. Specify the dimension of S. S={a a+d} where a,d are elements of R and V= M2x2 {a+d d } Homework Equations I guess I know the standard basis for M2x2 are the [(10 00) (01 00) (00 10) (00 01)]...
  30. radou

    What is the countable basis problem in topology?

    Homework Statement So, if X has a countable basis {Bn}, then every basis C for X contains a countable basis for X. The Attempt at a Solution First of all, consider all the intersections of elements of C of the form Ci\capCj. For every x in the intersection (if it's non empty), choose a...
  31. S

    Linear Algebra: basis and matrices

    Hi, I have a question about linear transformation. So given a matrix A in the basis u (denoted as A_u). Now in another basis that I don't know, A_u becomes A_v. How can I find v? (I know u, A_u and A_v). Thank you very much,
  32. N

    QM: Changing Basis | Why Use T_{σa,σb}?

    Homework Statement Hi Say I have the kinetic energy operator denoted by T(ri) for the particle i. I wish to represent it in some \left| \sigma \right\rangle -representation. My book says it is given by T = \sum\limits_{\sigma _a ,\sigma _b } {T_{\sigma _a ,\sigma _b } \left| {\psi...
  33. N

    Two-particle fermionic basis states

    Hi guys Say we are looking at a two-particle system consisting of two electrons (fermions). In my book it says that the basis states are given by \left| {\psi _{\alpha ,i} (r_m )} \right\rangle \left| {\psi _{\beta ,j} (r_n )} \right\rangle where rm and rn denote the two particles...
  34. F

    Solving Basis Functions Homework w/ Constants A_n & B_n

    Homework Statement Given x in the interval [0, \pi], let \phi_{0}(x) = 1, and \Phi_{n} (x) = sin ((2n-1)x). Show that there are constants: {A_{n}}^{n=0}_{\infty} and {B_{n}}^{n=0}_{\infty} such that: \sum^{n=0}_{\infty}A_{n}\phi_{n}=\sum^{n=0}_{\infty}B_{n}\phi_{n} But A_{n}...
  35. N

    Linear Transformations,Find basis of kernel and range

    suppose that vectors in R3 are denoted by 1*3 matrices, and define T:R4 to R3 by T9x,y,z,t)=(x-y+z+t,2x-2y+3z+4t,3x-3y+4z+5t).Find basis of kernel and range.
  36. B

    Finding the Basis of a Subspace in R^3 with Calculator | Linear Algebra Homework

    Homework Statement find the basis of a subspace of R^3 spanned by S: 1. S = { (4,4,8) (1,1,2) (1,1,1)} 2. S = { (1,2,2) (-1,0,0) (1,1,1) Homework Equations Im allowed to use calculator. The Attempt at a Solution Im not really sure what this is about. . .I tried the following...
  37. H

    What is the basis and dimension of the subspace U of P2?

    Homework Statement Find the basis and dimension of the following subspace U of P2 p(x) \ni P2 such that p(1) = p(2)Homework Equations The Attempt at a Solution I know all quadratics are in the form ax2 + bx + c set p(2) = p(1) 4a + 2b + c = a + b + c b = -3a Therefore ax2 -3a + c Basis(U)...
  38. B

    Unitary operators preserve normalization in arbitrary basis

    Homework Statement To test my knowledge of Sakurai, I asked myself to: "Prove that an operator being unitary is independent of basis." The Attempt at a Solution I want to show the expansion coefficients’ squared magnitudes sum to unity at time “t”, given that they do at time t = t0...
  39. P

    Time evolution operator on momentum basis

    Hey i was wondering how to express the time evolution operator U(t,to) to a momentum eigen state |p> for a particle moving in the xdirection under a zero potential, V= 0. The reason i need this is that iam told the only way to get the matrix element of the time evolution operator using position...
  40. J

    Gaussian basis set in DFT calculation

    I'm curious to know why chemists like to use Gaussian basis set in case of an ab-initio (ex.DFT) calculation. I understand that the molecules that are of interest to chemists are non-periodic and hence plane wave basis is not useful, but can't they use other real space basis like a grid? What...
  41. B

    Linear Algebra: A Basis for a Finite Dim VS

    Why is it enough to prove that a set of vectors is a BASIS to a FINITE DIMENSIONAL Vector Space, it is enough to show that it is Linearly Independent. No Need to prove that it spans the whole vector space?
  42. G

    Basis & Dimension: Subspace of R4

    Homework Statement Find a basis and dimension to each of the following subspaces of R4: U = {(a+b,a+c,b+c,a+b+c)|a,b,c∈R} Homework Equations The Attempt at a Solution I started by making a linear system. w(a + b) + x(a + c) + y(b + c) + z(a + b + c) = 0 a(w + x + z) + b(w...
  43. R

    Spin-1/2 Particle Transformation: Spin-Z to Spin-X

    I got a quick question about the transformation matrix from the spin-z basis to the spin-x basis for spin-1/2 particles. Would the matrix be: \left(\begin{array}{ccc} \frac{e^{i\theta}}{\sqrt{2}} &\frac{e^{i\delta}}{\sqrt{2}} \\ \frac{e^{i\theta}}{\sqrt{2}} & -\frac{e^{i\delta}}{\sqrt{2}}...
  44. mnb96

    Permutations of basis elements in Clifford Algebras

    Hello, let's consider, for example, the Clifford algebra CL(2,0) and the following mapping f for an arbitrary multivector: a + b\mathbf{e_1}+c\mathbf{e_2}+d\mathbf{e_{12}} \longmapsto a\mathbf{e_{12}} + b\mathbf{e_1}+c\mathbf{e_2}+d For vector spaces R^n we can permute the coordinates of...
  45. B

    Find Basis for Subspace: S with Degree ≤ 4 & f(0)=f(1)=0

    Homework Statement I need to find a basis for the following: S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0} 2. The attempt at a solution A general polymial is of the form: p(x) = ax^4 + bx^3 + cx^2 + dx + e Now for p(0) = p(1) = 0 I must have: e = 0 and a + b...
  46. F

    What is a Basis for a Polynomial Subspace with Specific Roots?

    Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...
  47. silvermane

    Finding a Transformation Matrix to yield the basis

    Homework Statement The solutions to the linear differential equation d^2u/dt^2 = u for a vector space. Find two independent solutions, to give a basis for that solution space. The Attempt at a Solution I want to understand this question. I feel that there's something I'm missing. I...
  48. P

    Linear Algebra: Basis Homework Help

    Homework Statement Homework Equations The Attempt at a Solution I don't know how to do this problem.. Help please!
  49. L

    Basis of Image of Linear Transformation

    Homework Statement Given a linear transformation F: R^3 --> R, F(x,y,z) = 3x-2y+z, find I am (F) and dim (Im (F)) Homework Equations I have found that dim(ker F) = 2 and from the theorem dim (V) = dim (Ker F) + dim (Im F), I know dim (V) = 3, so dim (Im F) = 1. The Attempt at...
  50. N

    Fourier of Basis Points (Basis in Reciprocal space) (Convolution Theorem)

    I came across this question where there is a FCC lattice. It states that the lattice is a convolution of the simple cubic (whose reciprocal lattice is itself) with a basis (that consists of 2 points). When finding the reciprocal of this BCC lattice, FourierTransform(BCC) =...
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