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. E

    The basis of n x n matrices with matrix multiplication

    Hi All, I recently came across the interesting notion of constructing the minimal set of nxn matrices that can be used as a basis to generate all nxn matrices given that matrix multiplication, and addition and multiplication by scalar are allowed. Is there a way to construct an explicit set...
  2. B

    Quadratic Forms: Closed Form from Values on Basis?

    Hi, Everyone: I have a quadratic form q, defined on Z<sup>4</sup> , and I know the value of q on each of the four basis vectors ( I know q is not linear, and there is a sort of "correction" for non-bilinearity between basis elements , whose values --on all pairs (a,b) of...
  3. P

    Understanding Change of Basis in Vector Spaces

    hi.. can anyone say what is the concept behind change of basis.. y do we change a vector of one basis to another?
  4. D

    QM: changing basis of quantum states

    Hello, I am trying to express a given wavefunction through different basis, momentum and position. Look at 5.2(b) and (c) through the link: http://qis.ucalgary.ca/quantech/443/2011/homework_five.pdf" I complete part (b) by doing the following...
  5. S

    Linear transformation with 2 ordered basis

    Homework Statement Suppose L:R^2 -> R^3 Find the matrix representing L(x) = Tx with respect to the ordered basis [u1,u2] and [b1.b2,b3] Homework Equations The Attempt at a Solution I've excluded the actual values since i can do the computation. Just wanted to make sure these steps are ok and...
  6. F

    Basis, what it really means for R

    Homework Statement Whenever LA talks about ℝn, do they mean just the n? Ex. Let's say I have two vectors \begin{bmatrix}1\\ 0\\ 0\end{bmatrix}\begin{bmatrix}0\\ 1\\ 0\end{bmatrix} Now it is tempting to say that the two vector is a basis for ℝ2. Now my professor tells me that it isn't a...
  7. I

    Vector Space Subspace Basis: Finding Compatible Bases

    Homework Statement Let S be a subspace of a vector space V. Let B be a basis for V. Is there a basis C for S such that C \subseteq B? not really sure how to approach this... any hints?
  8. G

    Understanding Change of Basis in N-Dimensional Space

    Homework Statement Could someone help me understand the following manipulations concerning change of babsis in an N-dimensional space: |i'\right\rangle=R|i\right\rangle=\sum_{j=1}^NR_{ji}|j\right\rangle multiply around by (R^{-1})_{ik}...
  9. D

    How to determine a basis given a set of vectors?

    Homework Statement Let V be the subspace spanned by the following vectors: [ 0]...[ 1 ]...[2] [ 2]...[ 1 ]...[5] [-1]...[3/4]...[0] Determine a basis for V. The Attempt at a Solution I'm not quite sure how to start here. Would placing the vectors in a matrix and deriving its...
  10. C

    What is the mathematical basis of quantum mechanics?

    Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.
  11. L

    How to Find an Orthonormal Basis in R3 Using Gram-Schmidt?

    Homework Statement Consider R3 together with the standard inner product. Let A = 1 1 −1 2 1 3 1 2 −6 (a) Use the Gram-Schmidt process to find an orthonormal basis S1 for null(A), and an orthonormal basis S2 for col(A). (b) Note that S = S1 ∪ S2 is a basis for R3. Use the the...
  12. Z

    Finding a Basis for P2(R): [2 + 5x + 4x^2]a = [1,2,3], etc

    Homework Statement If possible, find a basis a = {a1, a2, a3} of P2(R) such that... [2 + 5x + 4x^2]a = [1, 2, 3], [1 + x + x^2]a = [4,1,2] and [x + x^2]a = [3, -5, 1] 2. The attempt at a solution Basically, we have something like Ax = b for each of these, right? A* [2,5,4] =...
  13. B

    Basis for Margin of Error in Opinion Polls?

    Hi, Just curious as to what is the basis of the margin of error given in polls, e.g., in statements of the form:" 30% of people are in favor of candidate x. The poll has a margin of error of +/- 5 %" Thanks.
  14. I

    Comutational Chemistry - From Basis Set to MOs

    I am wondering if someone can explain exactly how we (or the computer more specifically) move from the basis set to the molecular orbitals. For example, If we use a 3-21G basis set this means the following: 1. We are approximating three slater type orbitals (STOs) using contracted...
  15. mnb96

    How to find a basis for the space of even functions (with some constraints)

    Hello, I am considering the set of all (differentiable) even functions with the following properties: 1) f(x)=f(-x) 2) f(0)=a_0, with a_0\in \mathbb{R} 3) f(n)=0, where n\in \mathbb{Z}-\{0\} One example of such a function is the sinc function sin(\pi x) / \pi x. Is it possible to find...
  16. M

    Orthonormal Basis Homework: Gram-Schmidt Process w/ Inner Product

    Homework Statement Hi, i am applying the gram-schmidt procedure to a basis of {1,2x,3x^2} with inner product <p,q> = \int p(x)q(x) from 0 to 1. i am unsure what to do with the inner product Homework Equations The Attempt at a Solution I have followed the procedure i have for...
  17. M

    Find Orthonormal Basis of R3: u1,u2,u3

    Homework Statement Note: the vectors are column vectors, not row vectors. Latex is not working for me right now. Find an orthonormal basis u1, u2, u3 of R3 such that span(u1) = span [1 2 3] and span(u1,u2) = span { [1 2 3], [1 1 -1] } Homework Equations The Attempt at...
  18. M

    Proving Quadratic Basis for P(2): t2+2t+1, t2+t, t2+1

    Prove that t2+2t+1,t2+t, t2+1 is a basis for the space of quadratic polynomials P(2).
  19. O

    Basis states, matrix elements and angular momentum

    Homework Statement The last 2 parts of the attached photo. (4 and 6 marks) Im really not sure how to go about them in a (clever) way that won't take 2 hours. Homework Equations Possibly the fact that the product of the raising/lowering operators, J-J+ = J2x + J2y Answers to previous...
  20. I

    Prove Basis to Basis Isomorphism

    Homework Statement Let t \in L(V,W). Prove that t is an isomorphism iff it carries a basis for V to a basis for W.Homework Equations L(V,W) is the set of all linear transformations from V to WThe Attempt at a Solution So I figured I would assume I have a transformation from a basis for V to a...
  21. R

    Find a basis for the space of 2x2 symmetric matrices

    a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis. b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.
  22. T

    Easy calculation of basis of the null space

    Homework Statement find the basis of the nullspace of this matrix \begin{pmatrix} 1&1&1&-1 \\ 0&0&1&3 \end{pmatrix}Homework Equations The Attempt at a Solution i forget it. i first substitue 0 and 1 for last row but what about the first row? Substiute 0 and 1 again and this will give 4 basis...
  23. M

    Finding basis of spaces and dimension

    Homework Statement Find a basis for each of the spaces and determine its dimension: The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0 Homework Equations The Attempt at a Solution So I jumped at this question without knowing too much about spaces and...
  24. T

    Linear Algebra: Image, Kernel and Basis

    Homework Statement I've been browsing the Internet but can't find a straightforward explanation for a procedure on how to find the image and kernel of a matrix. Question: Find a basis of the image of A, and a basis of the kernel of A. \[ A = \left[ {\begin{array}{ccc} 1 & 2 & 1 \\...
  25. A

    Relation between position of detector and basis of angular momentum

    Hi, I'm trying to understand if where I put my detector affects the basis in which I'm measuring. For example if I have a photon with spin | 1,1 \rangle _z (i.e. spin 1 in the z basis) emitted in the +z direction and I put a detector in the z direction I expect to see a circularly polarized...
  26. C

    Calculating Velocity and Distance in One-Dimensional Motion

    Homework Statement The mass of a van with a driver is 2000 kg . When the van accelerates, the velocity increases with a uniform acceleration of 3.0 m/s2. Homework Equations a) The van starts at rest. Find the velocity after 4.0 s. b) How far does the van travel in the first 4.0 s? c)...
  27. B

    Representation of a Matrix to a basis

    Homework Statement This problem refer to my previous post "trace of a matrix" M = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 1 & 5 \\ 0 & 5 & 3\end{pmatrix} from the following basis set: \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 &...
  28. P

    Change of Basis + Geometric, Algebraic Multiplicities

    Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues? I'm thinking that the answer is "no", it cannot.. Since if...
  29. B

    Finding a Basis Set for a Real Symmetrical 3x3 Matrix Space

    Let X denote the set of real symmetrical 3X3 matrix. Then (X,R) forms a linear space. What will be a basis set for this linear space? I would appreciate if someone can help me with the question. My understanding is in R3 space there could be many 3X3 matrix that could be the basis set for...
  30. mccoy1

    Linear Algebra: Change of basis

    (a) Let A (matrix) =c1= [1,2,1], c2 = [0,1,2], c3 = [3,-2,-1] be a matrix (c1,c2,c3 refer to the columns of the matrix A, which is a 3x3 matrix) expressed in the standard basis and let w1 = (0,0,1)T, w2 = (0,1,2)T , w3 =(3,0,2)T , find the vector AUE in w basis. (b). Referring to problem (a)...
  31. B

    Minimal elements of a MWI and the preferred basis problem

    Many physicists claim that decoherence determines the emergence of the worlds in the Many World Interpretation (MWI). I have always found such a claim elusively proved and actually wrong. Recently I wrote a paper: http://arxiv.org/abs/1008.3708 addressing such a subject, and I sent it to...
  32. A

    How to find a vector basis for this impossible form?

    Find a basis for the subspace S of R^3 consisting of all vectors of the form (a, 2a-b, b)^T, where a and b are real numbers Relevant equations would really just be the determinant of the system. I have tried so many 3x3 matrix combinations of the given form but no matter what the determinant...
  33. B

    Finding a Basis for Subspace U in Linear Algebra

    1. In each case, find a basis of the subspace U: (a) U=span{[1 -1 2 5 1].[3 1 4 2 7],[1 1 0 0 0],[5 1 6 7 8]} (b) U=span{[1 5 -6]^T, [2 6 -8]^T, [3 7 -10]^T, [4 8 12]^T} 2. Determine if the following sets of vectors are a basis of the indicated space: {[1 0 -2 5]^T,[4 4 -3 2]^T,[0 1...
  34. C

    Calculating Moles and Mass: Ca(NO3)2, Ca2+ ions, and NaCl - Homework Solutions

    Homework Statement a) What is the mass of one mole of Ca(NO3)2? b) How many Ca2+ ions are there in 0.05 moles of Ca(NO3)2? c) How many moles ofNaCl are there in 450 g of this substance? (Avogadro’s number is 6.022*1023 1/mol.) Homework Equations The Attempt at a Solution a)...
  35. S

    Basis of subspace (and combinations of them)

    Homework Statement We are given the following subspaces U := {x E R3: x1 + 2*x2 - x3 = 0} and V := {x E R3: x1 - 2*x2 - 2*x3 = 0} And we need to find a basis for (i) U (ii) V (iii) U n V (not an "n" but a symbol that looks like an upside-down U) (iv) span(U u V) (not a "u" but a symbol that...
  36. T

    Change of Basis Matrices for B1 and B2 in Vector Space V - Homework Solution

    Homework Statement Let B1 = {v1; v2; v3} be a basis of a vector space V and let B2 = {w1;w2;w3} where w1 = v2 + v3 ; w2 = v1 + v3 ; w3 = v1 + v2 Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2 and from B2 to B1. *Use the appropriate change of basis matrix...
  37. S

    Finding coefficients using dual basis

    Let V be the space of polynomials of degree 3 or less over \Re. For every \lambda\in\Re the evaluation at \lambda is the map ev_{\lambda} such that V \rightarrow \Re is linear. How do we find the coefficients of ev_{2} in the basis dual to \{1,x,x^2,x^3\}?
  38. N

    Bravais lattices and lattices with a basis

    Hi guys Ok, so one way to define a Bravais lattice is to say that each lattice point can be reached by R = la1+ma2+na3 for some integer m, l and n. Obviously, this cannot be the case when we have a lattice with a basis. But does that also mean that a lattice with a basis does not have...
  39. A

    Matrix Elements of Operators & Orthonormal Basis Sets

    So, the rule for finding the matrix elements of an operator is: \langle b_i|O|b_j\rangle Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R3) (finding matrix elements of linear...
  40. D

    Find a topological space which does not have a countable basis

    Homework Statement Find a topological space which does not have a countable basis. Homework Equations Definition of basis : A collection of subsets which satisfy: (i) union of every set equals the whole set (ii) any element from an intersection of two subsets is contained in another...
  41. T

    Topology: Clopen basis of a space

    Homework Statement So, I'm going through a proof and it is shamelessly asserted that the collection of clopen sets of {0,1}^{\mathbb{N}} is a countable basis. Can anyone reasure me of this, point me in the direction of proving it. Thanks Tal
  42. P

    Finding a Basis for a Reflection in R^2

    Find a basis Beta in R^2 such that the beta matrix B of the given linear transformation T is diagonal. The Reflection T about the line R^2 spanned by [1 2], [1 2] is suppose to be verticle. B=S^-1AS or B=[[T(v1)]beta [T(v20]beta] so i found the reflection matrix to be...
  43. E

    Every locally path connected space has a basis consisting of path connected sets

    Homework Statement The definition for local path connectedness is the following: let x be in X. Then for each open subset U of X such that x is in U, there exists an open V contained in U such that x is in V and the map induced by inclusion from the path components of V to the path components...
  44. A

    How can I show that trace is Invariant under the change of basis?

    How can I show that trace is Invariant under the change of basis?
  45. C

    Basis of skew symmetric matrix

    Homework Statement Let W be a 3x3 matrix where A^t(transpose)=-A. Find a basis for W. Homework Equations Find a basis for W. The Attempt at a Solution I have no idea how to start it.
  46. pellman

    Coordinate basis vs local frame?

    The wikipedia article on connection forms refers to a local frame. What is the relationship between local frames and coordinate bases? Are they the same thing? Is one a subset of the other? The connection form article uses general notation e_\alpha for the basis elements instead of the...
  47. D

    Is this proof for a topological basis ok?

    On the plane R^{2} let, B= {(a,b) x (c,d) \subset R^{2} | a < b, c < d } a.) Show that B is a basis for a topology on R^{2}. This means I have to show that every x in R^{2} is contained in a basis element, and that every point in the intersection of two basis elements is contained in...
  48. X

    Can every ideal be decomposed into a triangular Groebner basis?

    Groebner basis calculations are an important technique for solving systems of nonlinear equations. Despite being (in the worst case), computationally intractable, they seem to be effective solutions for a multitude of problems. I have a problem that I am trying to solve. I have perused the...
  49. D

    Topology of R: Basis and Rationals

    Consider the collection of sets C = {[a,b), | a<b, and and b are rational } a.) Show that C is a basis for a topology on R. b.) prove that the topology generated by C is not the standard topology on R.So, I know for C to be a basis, there must be some x \in R, and in the union of some C1...
  50. Q

    Change of Basis Homework: Solving System of Equations

    Homework Statement We are given 2 bases for V = \Re_{1 x 3}. They are \beta_{1} = \begin{bmatrix} 2 & 3 & 2\end{bmatrix} \beta_{2} = \begin{bmatrix} 7 & 10 & 6\end{bmatrix} \beta_{3} = \begin{bmatrix} 6 & 10 & 7\end{bmatrix} and, \delta_{1} = \begin{bmatrix} 1 & 1 &...
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