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. F

    LINEAR ALGEBRA: image of vectors through other basis

    Homework Statement In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)## and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where (1 0 0) (0 2 0) = A (0 0 0)...
  2. O

    Finding Basis & Spanning Set for Matrix: a,b,c,d

    I'm having trouble finding the spanning set and basis for the matrix; | a b | | c d | with condition that b=d I'm thinking thinking the spanning set would be A= x B = y C = z Such that x,y,z are all reals, but I can't think of how to find a basis for this, I'm thinking of doing...
  3. marcus

    New basis for atoms of spatial geometry (intertwiners)

    Here's the introduction of the paper by Freidel and Hnybida. Quantum geometry is built up of chunks of geometry that contain information relating to volume, areas, angles made with neighbor chunks, etc. The Hilbert space that these chunks (called intertwiners) live in needs a set of basis...
  4. Ravi Mohan

    Particle in a box in momentum basis

    I have been thinking about it for sometime but couldn't really get the answer. This is the progress I have made till now. E |ψ> = H |ψ> E <p|ψ> = <p|H|ψ> Now how to evaluate the number <p|H|ψ>? Although I can evaluate this number by introducng identity operator 1 = \int|x><x| dx. But for...
  5. S

    Basis of Vectors: Definition & Examples

    So, I know that for a set of vectors to be a basis the set of vectors must be linearly independent and also must be a spanning set of vectors. So, they can't be parallel. I still feel that I'm not fully understanding what a basis is. Could someone explain to me, maybe with an example, what is a...
  6. L

    Find the basis for both eigenvalues

    Homework Statement Given matrix A= {[39/25,48/25],[48/25,11/25]} find the basis for both eigenvalues. Homework Equations The Attempt at a Solution I row reduced the matrix and found both eigenvalues. I found λ = -1, and λ = 3. Then, I used diagonalization method [-1I2 - A 0]...
  7. O

    Linear Transformation using Two Basis

    Hi, I'm having trouble understanding the purpose of using two basis in a linear transformation. My lecturer explained that it was a way to find a linear transformation that satisfied either dimension, but I'm having trouble understanding how that relates to the method in finding this...
  8. R

    What's the difference between lattice vectors and basis vectors?

    Google has not been very useful, and Kittel has too little on crystallography. Actually, what's a good source on crystallography?
  9. F

    Linear algebra: orthonormal basis

    Homework Statement ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix (1 0 0) (0 2 0) =##M_{\phi}^{B,B}##= (0 0 3) where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1)) Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi## The...
  10. D

    Write Sz in the J angular momentum basis?

    I'm working on a problem where I want to write the operator S_z down in terms of some operator(s) in the \vec{J} = \vec{L} + \vec{S} basis so that I can operate S_z on the states \mid \ell, s=1/2, j= \ell\pm1/2, m\rangle but I'm having trouble finding the correct combination of operators...
  11. K

    MHB Exploring Basis and Vectors in Linear Algebra

    Hi guys, I'm back and have another Linear Albgera question! Thanks in advance. No idea how to start
  12. E

    Physical observables, locality, and a preferred basis

    Quantum mechanics says that physical observables are self-adjoint operators. Is this correspondence a bijection, ie can we realize any such operator as a physical observable? There are obvious practical concerns with physically realizing certain contrived operators. But are there any...
  13. M

    User-defined orthonormal basis

    Does anybody know how to create a orthonormal basis, i.e. a matrix containing orthogonal vectors of norm 1, out of a given direction (normalised vector or versor) in a space with dimension N>3? With "out of a given direction", I mean that the resulting basis would have the first vector equal...
  14. U

    Finding an orthonormal basis of V

    I've done most of this question apart from the very last bit. I have an answer to the very last bit, but it doesn't use any of my previously proved statements and I think they probably mean me to deduce from what I already have. Homework Statement Let V be the finite-dimensional vector...
  15. V

    Orthonormal basis vectors for polar coordinate system

    Firstly; is there a difference between the "regular" polar coordinates that use \theta and r to describe a point (the one where the point (\sqrt{2}, \frac{\pi}{4}) equals (1, 1) in rectangular coordinates) and the ones that use the orthonormal basis vectors \hat{e}_r and...
  16. B

    Quick Quantum Mechanics Q about basis

    Homework Statement Suppose we have a system and that {|a>, |b>, ...} is a complete and orthonormal basis for the system Am i right in thinking Ʃ(j) <k|j><j|i> = <k|i> = 0 unless k=i? In other words, does the LHS expression equal the middle one because Ʃ(j) |j><j| is just the insertion...
  17. K

    Finding a Basis for Subspace a of R^3

    Homework Statement Find bases for the following subspace a of r^3 Y+z=0 The Attempt at a Solution First I found a normal to this plane n=(0,1,1) Then I found two vectors which are orthogonal to the normal u=(0,-1,1), v=(1,0,0) Is this correct the answer in my book has...
  18. T

    Operator Transformation under Change of Basis

    Homework Statement Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...
  19. T

    Probability of Finding System in a State Given a Particular Basis

    Homework Statement Note: I am going to use |a> <a| to denote ket and bra vectors The components of the state of a system| ω1> in some basis |δ1>, |δ2>, |δ3> are given by <δ1|ω1> = i/sqrt(3), <δ2|ω1> = sqrt(2/3), <δ3|ω1> = 0 Find the probability of finding the system in the state |ω2>...
  20. T

    Complex Number Orthonormal Basis.

    Find an orthonormal basis for P2(ℂ) with respect to the inner product: <p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer Attempt: This is where I'm having trouble. So usually I'm given...
  21. R

    How Do You Convert Position to Momentum Space in Quantum Mechanics?

    Homework Statement write the following in K basis: A=∫|x><x|dx where the integral limits are from -a to a Homework Equations The Attempt at a Solution I tried solving it by inserting the identity I=∫|k><k|dk where the integral limits are from -∞ to +∞ but then I do not...
  22. B

    Finding basis of 3x3 matrix space

    Homework Statement For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first...
  23. A

    Expectation value in unpertubed basis

    I have a question regarding an exercise I am doing. It is an electron confined to move on a cylinder and I am asked to: "Find the expectation value of Ly and Lz" in the unperturbed basis. I am just not sure what is meant by the expectation value in a basis? I know what the expectation value is...
  24. Fernando Revilla

    MHB Matteo's question at Yahoo Answers (change of basis)

    Here is the question: Here is a link to the question: Matrix of change of basis? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  25. C

    Understanding Coordinate Basis in Wald's GR

    In Wald's GR he makes use of a coordinate basis consisting of ∂/∂x^{n} where n runs over the coordinates, and I understand his argument that ∂f/∂x^{n} are tangent vectors, but I can't wrap my head around the operator ∂/x^{n} spanning a tangent space of a manifold. Any clarification on this would...
  26. mnb96

    Orthonormal basis functions for L^2(R)

    Hello, are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)? According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of...
  27. S

    Measuring a qubit in a different basis

    Hi, I'm a student of computer science and am writing a simulator for Measurement-based quantum computing or one-way quantum computing (it's based on the paper "The Measurement Calculus" by Elham Kashefi et al.). Anyway, I'm still a bit confused when it comes to the calculations. If we write...
  28. Fernando Revilla

    MHB Matrix representation of T with Basis B?

    I quote un unsolved question posted in MHF on December 8th, 2012 by user bonfire09.
  29. T

    Finding a basis for a subspace of Z.

    Find a basis for the subspace S = span{(1,2,1,2,1) , (1,1,2,2,1), (0,1,2,0,2)} of Z53 (The set of elements in the field of modulus 3) Attemept: So the issue isn't in finding a basis per say. If this was the field of Real numbers I wouldn't have an issue, I would just row reduce and use the...
  30. M

    MHB Exploring the Relationships between Triangles - A Mathematical Analysis

    See a picture that represents the relations of the two triangles what is a "?" 3?3=3 3?3=4 3?3=5 3?3=6 3?3=7 3?3=8 3?3=9 3?3=10 3?3=12
  31. S

    Can You Explain the Relationship Between a Basis and Its Dual Basis?

    Hi, I'm learning about vector spaces and I would like to ask some questions about it. Suppose I have a vector space V, and a basis for V \{v_1, ... v_n\}. Then there is a dual space V^* consisting all linear functions whose domain is V and range is ℝ. Then the space V^* has a dual basis \{x_1...
  32. C

    How Does a Two Atoms Basis Lattice Act Independently?

    Homework Statement For a lattice with a two atoms basis, the two dispersion relations valid for Ka = ±∏ w2 = 2C/M2 and w2 = 2C/M1 Show that under these conditions the lattice acts as two independent lattices (one lattice per each atom) with one of the lattices moving while the other is...
  33. T

    The tension between addition and multiplication - the basis for all mathematics?

    Throughout mathematics, we see the appearance of pi and e, they are integral to every part of mathematics. could this be because the reason all or at least most mathematical problems arise because of the tension between multiplication and addition (pi and e) itself?. certainly the deepest...
  34. C

    Problem with changing basis in Hilbert space

    The expansion theorem in quantum mechanics states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator. If that's the case then that would imply we would be able to represent the spin state of a particle in terms of...
  35. N

    How can matrices be used as a basis for linear mappings?

    Hi. Define a linear mapping F: M2-->M2 by F(X)=AX-XA for a matrix A, and find a basis for the nullspace and the vectorspace(not sure if this is the term in english). Then I want to show that dim N(F)=dim V(F)=2 for all A, A≠λI, for some real λ. F(A)=F(E)=0, so A and E belongs to the nullspace...
  36. R

    What Generators Can Run on 12 kgm Torque and Their Current Output?

    My project produces 12 kgm torque so,I need a list of Generator which runs on 12 kgm torque and how much Current it produces... Please advise
  37. TrickyDicky

    Fourier transform as (continuous) change of basis

    Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...
  38. B

    How does the change of basis theorem work in linear algebra?

    Let B={b1,b2} and C={c1,c2} be basis. Then the change of coordinate matrix P(C to B) involves the C-coordinate vectors of b1 and b2. Let [b1]c=[x1] and [b2]c=[y1] ...[x2]...[y2]. Then by definition [c1 c2][x1]=b1 and [c1 c2][y1]=b2. I don't get how you can ....... [x2].....[y2] multiply the...
  39. G

    Matrix Representation of Operators in a Finite Basis

    Homework Statement I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text. Let's say you have a wave function of the following form for a linear harmonic oscillator: \Psi = c_1 | E_1 \rangle + c_2 | E_2...
  40. B

    Linear Mapping T: P2 to P2 with Basis B | Homework Help & Solution Explained"

    Homework Statement Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}. The part that I am confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't...
  41. O

    How to write basis for symmetric nxn matrices

    Homework Statement Write down a basis for the space of nxn symmetric matrices. The Attempt at a Solution I just need to know what the notation for this sort of thing is. I understand what the basis looks like, and I was even able to calculate that it would have dimension...
  42. C

    Set of vectors with each subset forming a basis

    Hey! Let M and N be two natural numbers and N>M. I want to build a set A with N vectors of size M such that each subset S of A, where |S| = M, contains linearly independent vectors. Another way to put it is that every S should be a basis for R^M. Any ideas? Thanks!
  43. B

    Change of basis of density matrix

    I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this? Any help really appreciated- thanks!
  44. I

    Using the rotation operator to solve for eigenstates upon a general basis

    Homework Statement I need to express the rotation operator as follows R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2) given the fact that R(uj)= e^(iuS_y/(\hbar)) using |+-z> as a basis, expanding R in a taylor series express S_y^2 as a matrix Homework Equations I know...
  45. K

    Orthogonalizing a basis by gram schmidt process

    Homework Statement (a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}. (b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations. Homework Equations gram schmidt orthogonalization and...
  46. shounakbhatta

    Understanding Change of Basis Vector: What it is and How to Use It

    Hello, I am doing calculation on change of basis vector. But I am unable to understand why we do it. I mean to say what is the use of it and where in physics or maths it is used. Can anybody please explain it?
  47. M

    News Is there any scientific basis for Proposition 37?

    I'm sure you've all heard of Prop 37, but I'll write a short introduction. In the state of California, which is located in the United States, residents can vote on a proposition. That proposition becomes law if they vote in favor of it. Proposition 37 was created in response to the belief...
  48. matqkks

    Transform Vector: Real Life Examples

    Why would we want to transform a vector in our normal basis (xyz axes) to another basis? The only situation I can recall is when we are looking at a force applied on an inclined plane. Are there any other real life examples where this may be necessary?
  49. S

    How unitary change of basis related to Trace?

    Homework Statement Shanker 1.7.1 3.)Show that the trace of an operator is unaffected by a unitary change of basis (Equivalently, show TrΩ=TrU^{\dagger}ΩUHomework Equations I can show that via Shanker's hint, but I however can't see how a unitary change of basis links to TrΩ=TrU^{\dagger}ΩU...
  50. P

    Basis, Nullspace, Linear transformtion

    The problem is attached, I did parts 1-3, but I am having trouble with part 4. This is what i was planning on doing for part 4 (my teacher said this wasn't the correct method): set T(v)=0 and solve the augmented matrix 1 0 -1 1 0 2 1 -2 4 0 3 1 -1 7 0 rref gives 1 0 0 2 0 0 1 0 2 0 0 0 1 1 0...
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