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

    Why Does the Second Equality Hold in Multi-Linear Algebra?

    {(a_i)_j} is the dual basis to the basis {(e_i)_j} I want to show that ((a_i)_1) \wedge (a_i)_2 \wedge... \wedge (a_i)_n ((e_i)_1,(e_i)_2,...,(e_i)_n) = 1 this is exercise 4.1(a) from Spivak. So my approach was: \BigWedge_ L=1^k (a_i)_L ((e_i)_1,...,(e_i)_n) = k! Alt(\BigCross_L=1^k...
  2. K

    Pseudoinverse - change of basis?

    Hello, I was wondering if the pseudoinverse can be considered a change of basis? If an m x n matrix with m < n and rank m and you wish to solve the system Ax = b, the solution would hold an infinite number of solutions; hence you form the pseudoinverse by A^T(A*A^T)^-1 and solve for x to...
  3. H

    Find a kernel and image basis of a linear transformation

    Homework Statement Find a kernel and image basis of the linear transformation having: \displaystyle T:{{\mathbb{R}}^{3}}\to {{\mathbb{R}}^{3}} so that \displaystyle _{B}{{\left( T \right)}_{B}}=\left( \begin{matrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 0 & 0 & 0 \\ \end{matrix} \right)...
  4. S

    Understanding the relevance of writing a quantum state in the Schmidt basis

    Hi all, This is both linear algebra and physics problem, and I decided to post in physics because I want a "physics-framed" answer. Suppose you have a system with two objects (subsystems) in it described by the state: |ψ> = ƩiƩjcij|i>|j> where |i> and |j> are orthonormal bases for the two...
  5. T

    If V is a 3-dimensional Lie algebra with basis vectors E,F,G

    If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...
  6. L

    Does every independent set spans the space necessarily form a basis?

    Hi there! A Hilbert space E is spanned by a set S if E is generated by the element of S. It is well known that in the finite dimensional case that S spans E and S is linearly independent set iff the set S form a basis for E. The question is that true for the infinite dimensional...
  7. H

    Orthogonal Basis for a subspace

    Homework Statement Let W = \begin{cases} \begin{pmatrix}x\\y\\z\\w\end{pmatrix} \in R^4 | w + 2x + 2y + 4z = 0 \end{cases} A)Find basis for W. B)Find basis for W^{\perp} C)Use parts (A) and (B) to find an orthogonal basis for R^4 with respect to the Euclidean inner product. Homework...
  8. B

    Circular coordinate space using an orthonormal basis

    If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was...
  9. H

    Finding a matrix with respect to standard basis

    Homework Statement Homework Equations ... The Attempt at a Solution Can someone just point me how to approach this? Do we take a random second degree polynomial and input 2x + 3 instead of x, then find the constants (eg. denoted by a , b , c) by putting the new equation equal to...
  10. I

    Subspace as a Direct Sum of Intersections with Basis Partition?

    I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...
  11. H

    Basis and Dimension of a Subpace.

    Homework Statement V = {p(x) belongs to P3 such that p'(1) + p'(-1) = 0} Homework Equations ... The Attempt at a Solution Okay, so finding the first derivative of p(x) = ax^3 + bx^2 + cx + d and plugging in the values 1 and -1 (to find p'(1) and p'(-1)), we get c = -3a. Does this make the...
  12. J

    Finding a basis in ImT using Gaussian Elimination

    Homework Statement $$ \begin{pmatrix} -1&3&0\\ 2&0&-1\\ 0&-6&1 \end{pmatrix} $$ Finding the ImT basis of this The Attempt at a Solution I got it down to $$ \begin{pmatrix} 1&0&-1/2\\ 0&1&1/6\\ 0&0&1 \end{pmatrix} $$ I know that by the principle of having pivots as the only non-zero...
  13. S

    Undergraduate Mechanics (Problem with force expressed as basis vectors)

    Homework Statement The problem along with its solution is attached as Problem 1-2.jpg. Homework Equations Norm of a vector. The Attempt at a Solution Starting from the final answer of the solution, sqrt((-0.625)^2 + (0.333)^2) == 0.708176532 != 1. Did the book do something wrong? I ask...
  14. A

    Finding basis of a column space/row space

    I was wondering whether we can use row as well as column operations to reduce a matrix to find column space? Or do we only have to perform row operations to reduce matrix in case of row space and column operations to find column space?
  15. R

    Determine whether the given vectors form a basis

    Homework Statement w1 = 2 1 2 w2 = 1 -2 -3 w3 = 5 0 1 for R3 Homework Equations The Attempt at a Solution The books says the above is not a basis, why not? There are no free variables, none of the vectors are multiples of the other, they are linearly independent and the...
  16. A

    Determinats,dependence, span, basis.

    Im having trouble under stand the relationships between determinats, span, basis. Given a 3x3 matrix on R3 vector space. * If determinat is 0, it is linearly dependent, will NOT span R3, is NOT a basis of R3. , If determinant is non-zero, its linearly independent, will span R3, is a basis of...
  17. K

    How can a basis in a vector space be used to determine linear independence?

    From what I understand, a basis is essentially a subset of a vector space over a given field. Now what I'm not so sure of is the linearly independence part. If the basis has two linearly independent vectors, then than means they aren't collinear: rather, they wouldn't have the same slope...
  18. B

    Derivative of transformation of basis in R^n

    I've just studied the implicit function theorem and if we assume the theorem is true then we can easily compute the following: id_n = D(id_n)_x = D({f^-1}°{f})_x = {D(f^-1)_f(x)} ° {Df_x} where D(*)_a means the derivative of * at x. OKay... so this was very straightforward until I began...
  19. R

    MHB Find polynomials in S, then find basis for ideal (S)

    Hi There, I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a...
  20. D

    Find if 3 vectors form a basis in space R^4

    Homework Statement Is the system of vectors a1=(1,-1,0,1), a2=(2,3,-1,0), a3=(4,1,-1,4) linearly independent? Do these vectors form a basis in the vector space R^4? State why. Homework Equations The Attempt at a Solution I have done the first part of the exercise. I have found...
  21. M

    Basis, Linear Transformation, and Powers of a Matrix

    Homework Statement Let A be an 3x3 matrix so that A^3 = {3x3 zero matrix}. Assume there is a vector v with [A^2][v] ≠ {zero vector}. (a) Prove that B = {v; Av; [A^2]v} is a basis. (b) Let T be the linear transformation represented by A in the stan- dard basis. What is [T]B? Homework...
  22. V

    Capturing n basis vectors by single one

    Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in Fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just...
  23. B

    Basis vectors and abstract index notation

    First of all, I'd like to say hi to all the peole here on the forum! Now to my question: When reading some general relativity articles, I came upon this strange notation: T^{a}_{b} = C(dt)^{a}(∂_{t})_{b} + D(∂_{t})^{a}(dt)_{b}. Can someone please explain to me what this means? Clearly...
  24. K

    About basis of the honeycomb lattice

    Hi there, I am reading the book "Condensed Matter Physics" second edition by Michael P. Marder. It stated in page 9 that one basis of the the honeycomb lattice is \vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})] which is based on figure 1.5(B) in page...
  25. K

    Finding a basis for the following vectors.

    Find a basis and the dimension for the subspace of R^3(3D) spanned by the vectors {(0,1,-2),(3,0,1),(3,2,-3)} The dimension is 2 regardless if i put the vectors in row space or column space form. But to find the basis I need to put it in row space form. Can anyone please explain when I...
  26. K

    Finding the basis for a set of vectors.

    The set is (1,3),(-1,2),(7,6) it is in R2 so I don't get why there are 3 elements. I assumed they are not vectors but points instead. but if they are points then it becomes a line, and the answer is that its dimension is 2, and a basis is (1,0) and (0,1) Could someone explain this? Thanks
  27. S

    Basis of kernel and image of a linear transformation. (All worked out)

    http://dl.dropbox.com/u/33103477/linear%20transformations.png My solution(Ignore part (a), this part (b) only) http://dl.dropbox.com/u/33103477/1.jpg http://dl.dropbox.com/u/33103477/2.jpg So I have worked out the basis and for the kernel of L1 and image of L2, so I have U1 and U2...
  28. S

    Procedure for orking out the basis of the kernel of a linear transformation.

    I am working on a problem dealing with transformations of a vector and finding the basis of its kernel. Now I have worked out everything below but after reading the definitions I am a bit confused, hence just want verification if the procedure I am following is correct. My transformed matrix is...
  29. S

    Basis of linear transformations

    http://dl.dropbox.com/u/33103477/linear%20transformations.png My attempt was to first find the transformed matrices L1 and L2. L1= ---[3 1 2 -1] -------[2 4 1 -1] L2= ---[1 -1] -------[1 -3] -------[2 -8] -------[3 -27] Now reducing L1, I have -------[1 0 7/10 -3/10]...
  30. S

    A one-form versus a dual basis vector

    Hi everyone, Pardon the neophyte question, but is a one-form the same thing as a dual basis vector? If not, are they related in some way, or completely different concepts/entities? Thank you!
  31. A

    How to find the orthogonal basis?

    Can somebody help me how to approach this problem.I am having trouble finding the orthogonal basis.
  32. E

    Taking vectors from one basis to another [Byron and Fuller]

    B&F have the following: \delta_{i j} = e'_i \cdot e'_j = a_{i k} \left( e_k \cdot e'_j \right) = a_{i k} a_{j k} and they ask the reader to show that a_{k i} a_{k j} = \delta_{i j} Does it suffice to show the following? : \delta_{i j} = a_{i k} a_{j k} \to \delta_{j i} = \left(...
  33. S

    Difficult theoretical problem on basis vectors

    How the hell do you prove that the components of a vector w.r.t. a given basis are unique? I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!
  34. V

    Need for Separate Basis for Kernel: Explained by Hello

    hello :) I was trying to prove the following result : for a linear mapping L: V --> W dimension of a domain V = dimension of I am (L) + dimension of kernel (L) So, my doubt actually is that do we really need a separate basis for the kernel ? Theoretically, the kernel is a subspace of the...
  35. T

    What is the basis for W perp in the Gram Schmidt process?

    Homework Statement Let W be a subspace of ℝ4 spanned by the vectors: u1 = [1; -4; 0; 1], u2 = [7; -7; -4; 1] Find an orthogonal basis for W by performing the Gram Schmidt proces to there vectors. Find a basis for W perp (W with the upside down T). Homework Equations Gram...
  36. R

    Basis functions of a differential equation, given boundary conditions

    First off, I've never taken a differential equations class. This is for my Math Methods for Physicists class, and we are on the topic of DE. Unfortunately, we didn't cover this much, so most of what I am about to show you comes from the professor giving me tips and my own common sense. I'd...
  37. F

    Linear transformation with standard basis

    Homework Statement Let s be the linear transformation s: P2→ R^3 ( P2 is polynomial of degree 2 or less) a+bx→(a,b,a+b) find the matrix of s and the matrix of tos with respect to the standard basis for the domain P2 and the standard basis for the codomain R^3 The Attempt at a...
  38. B

    How Does the Preferred Basis Problem Impact Quantum State Representation?

    Homework Statement I'm trying to understand the preferred basis problem in the foundations of QM Ok so I read somewhere that in general any state can be decomposed in different ways. I don't quite see how this is meant to work Suppose 'up' / 'down' represent z component of ang mom...
  39. S

    Finding a basis given part of that basis

    Homework Statement This is the question on my assignment: In each case below, given a vector space V , find a basis B for V containing the linearly independent set S ⊂ B. It has a bunch of different cases but I think that if you help me with the following two, I will learn enough to do...
  40. T

    Is Change of Basis the Solution to My Linear Algebra Problem?

    Homework Statement The Attempt at a Solution So first I thought to myself that the proper way of doing this problem was to construct each of the standard basis vectors as a linear combination of the basis given us. I have, T(1,0,0) = \frac{1}{2} T(1,0,1) + \frac{1}{2} T(1,0,-1) =...
  41. F

    Jordan Basis for Differential Operator

    Homework Statement Let V = P_n(\textbf{F}) . Prove the differential operator D is nilpotent and find a Jordan basis. Homework Equations D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} The Attempt at a Solution I already did the proof of D being nilpotent, which was easy. But we haven't covered...
  42. F

    Gram-Schmidt Method for orthogonal basis

    I have S= {(1,1,0,1) (1,0,-1,0) (1,1,0,2)} its one of the subset and second it T= {(x,y,z,2x-y+3z)} If you were to use Gram-Schmidt method to find the orthogoan basis for T who would you processed? I really don't understand this concept. I know from T , the hyperplane is 2x-y+3z so the...
  43. M

    Reduced Grobner basis form a regular sequence?

    Does anyone know if a set of homogeneous polynomials forms a reduced Grobner basis, then they form a regular sequence in the polynomial ring? Any references? All the references that I have looked at (so far) have not related the two. If this is not true, can you give me a counterexample...
  44. M

    Can an electron move to a higher energy level on a permanent basis?

    Hi, I have some very basic questions regarding electron energy levels/states. In the basic atom model when an electron becomes excited (i.e. absorbs a photon or collides with a nearby atom or particle) and moves into an energy state greater than its ground state, must it always eventually...
  45. T

    Orthogonal Basis and Inner Products

    Homework Statement The Attempt at a SolutionSince A is a vector in V and since the A_i form a basis, we can write A as a linear combination of the A_i. We write A = x_1 A_1 + ... + x_n A_n. Thus, we have, <x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i>. Because...
  46. F

    Crystals: difference between basis and unit cell

    Hello Forum, a lattice is a set of points. We can place a basis at each set of points. The basis can be one atom or a group of atoms. I thought that a translation of the basis would produce the whole crystal... How is a basis different from the unit cell? Are they the same thing...
  47. A

    Trouble understanding MTW particulary noncoordinate basis

    Background:(you might not be interested so you can skip if you want) I am trying to learn general relativity using the Book Gravitation by Misner, Thorn and Wheeler. The book for the most part seems easy for me to understand but once in a while words i neither heared nor can find the meaning of...
  48. F

    Density matrix elements, momentum basis, second quantization

    Hello everyone, I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise. Specifically, I'm having a hard time understanding the matrix elements of the density matrix...
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