What is Basis: Definition and 1000 Discussions

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. H

    Proving Equivalence of Standard and Basis-Generated Topologies on RxR

    I am having trouble proving this statement. Please help as I am trying to study for my exam, which is tomorrow Prove that the standard topology on RxR is equivalent to the one generated by the basis consisting of open disks. Thanks :)
  2. Z

    Polynomial Basis and Linear Transformation

    Homework Statement Let X be the vector space of polynomial of order less than or equal to M a) Show that the set B={1,x,...,x^M} is a basis vector b) Consider the mapping T from X to X defined as: f(x)= Tg(x) = d/dx g(x) i) Show T is linear ii) derive a matrix...
  3. Z

    Find Basis of R^n Subspace with Sum X^2=1

    Homework Statement Let V=(x1,x2,..Xn) Sum X = 0 be a subspace of R^n. Find a basis of V such that Sum X^2=1 Homework Equations The Attempt at a Solution for Sum X =0 x1+x2+..+xn =0 xn= -x1-x2-... So <x1, x2, ...,-x1-x2-...>=<x1, 0,0...,-x1> +<0,x2,0,...,-x2)+...
  4. Somefantastik

    Basis functions for polynomial

    Homework Statement For I = [a,b], define: P3(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a3x3 + a2x2 + a1x + a0}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b). Homework Equations The Attempt at a Solution I'm not exactly sure...
  5. D

    2nd ODE, Reduction of Order, Basis known

    I have a homework problem where I am to find y_2 for a 2nd ODE, with y_1=x. I'm familiar with the process of: let y_2 = ux y_2- = u'x u y_2'' = 2u' + u''x substituting these terms into the 2ODE, then letting u' = v. When integrating v and u' to solve for u, do I need to include...
  6. I

    Change of basis in R^n and dimension is <n

    Suppose I have a basis for a subspace V in \mathbb{R}^{4}: \mathbf{v_{1}}=[1, 3, 5, 7]^{T} \mathbf{v_{2}}=[2, 4, 6, 8]^{T} \mathbf{v_{3}}=[3, 3, 4, 4]^{T} V has dimension 3, but is in \mathbb{R}^{4}. How would one switch basis for this subspace, when you can't use an invertible...
  7. F

    Physical basis of Neurophysiology

    Any suggestions on how to go about becoming a researcher in this area? I have very little formal background in math and physics and will be studying this on my spare time (which means when I'm not working on the cognitive stuff, though I would love to integrate something more rigorous into...
  8. C

    KG and SEC as the basis of all units?

    About 6 minutes ago I thought of this, and I want to check if it is true. Is the kilogram and the second a basis for all units in existence? That is, can all units be derived from these two? I can't think of any other units that are independent.
  9. L

    Physical meaning of orthonormal basis

    I want to know what orthonormal basis or transformation physically means. Can anyone please explain me with a practical example? I prefer examples as to where it is put to use practically rather than examples with just numbers..
  10. A

    Finding Orthonormal Basis of Hilbert Space wrt Lattice of Subspaces

    I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...
  11. K

    Representation of angular momentum matrix in Cartesian and spherical basis

    The two sets of matrices: {G_1} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ \end{array}} \right){\rm{ }}{G_2} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 1 \\ 0 & 0 & 0 \\ { - 1} & 0 & 0 \\ \end{array}} \right){\rm{...
  12. B

    Hamel basis and infinite-dimensional vector spaces

    If we could find the Hamel basis for any infinite dimensional vector space, what kind of consequences would this have?
  13. J

    Proof that gamma matrices form a complete basis

    Hi all, I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for 4\times4 matrices. In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various...
  14. jinksys

    Linear Algebra - Basis for a row space

    A = 1 2 -1 3 3 5 2 0 0 1 2 1 -1 0 -2 7 Problem: Find a basis for the row space of A consisting of vectors that are row vector of A. My attempt: I transpose the matrix A and put it into reduced row echelon form. It turns out that there are leading ones in every column...
  15. jinksys

    Lin Alg - Find the basis and dimension

    Find the basis and dimension of the following homogeneous system: A = |1 0 2| |x1| |2 1 3| |x2| = [0,0,0] |3 1 2| |x3| My attempt: Solving the coefficient matrix for RREF, I get the identify matrix. So, x1=x2=x3=0 and the only solution is a trivial one. Does that mean...
  16. jinksys

    Lin Algebra - Find a basis for the given subspaces

    Find a basis for the given subspaces of R3 and R4. a) All vectors of the form (a, b, c) where a =0. My attempt: I know that I need to find vectors that are linearly independent and satisfy the given restrictions, so... (0, 1, 1) and (0, 0, 1) The vectors aren't scalar multiples of...
  17. jinksys

    Linear Algebra - Number of vectors in a basis

    As I read one linear algebra book I have, I am told that "If a vector space V has a basis with 'n' vectors, then every basis in vector space V has 'n' vectors. So every basis in R3 has 3, every basis in R4 has 4, etc. However, I have a problem that says: Let S = { "five vectors" } be a...
  18. T

    If (a,b,c) is a Basis of R3 , Does (a+b,b+c,c+a) also a Basis of R3

    Homework Statement Well the same as the subject.. (a,b,c) is a Basis of R^{3} does (a+b , b+c , c+a) Basis to R^{3} I have another question .. is (a-b , b-c , c-a) Basis to R^{3} This is know is not true because if I use e1, e2 , e3 I got a error line. Homework Equations start of...
  19. K

    Linear Algebra: Orthogonal basis ERG HELP

    Homework Statement Consider the vector V= [1 2 3 4]' in R4, find a basis of the subspace of R4 consisting of all vectors perpendicular to V. Homework Equations I mean, I'm just completely stumped by this one. I know that in R2, any V can be broken down to VParallel + VPerp, which...
  20. F

    Basis and dimension: Need help finding my mistakes

    Homework Statement He made some notes, but I'm still confused. The Attempt at a Solution
  21. K

    Magnetic field produced by a short wire element (on 360 degree basis)

    In consideration of a lone wire element of differential length dL carrying a current I: Two things concern me here: 1) Is a magnetic field generated by current I really limited to only spaces that exist orthogonally to the line between the two ends of the wire element? If not, what does...
  22. G

    Dual basis problem. (Linear Algebra)

    Homework Statement Prove that if m < n and if y_1,...,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j] = 0 for j = 1,..., m Homework Equations The Attempt at a Solution My thinking is somehow that we...
  23. W

    Why the basis of the tangent space of a manifold is some partials?

    it is quite peculiar i know you do not want to embed the manifold into a R^n Euclidean space but still it is too peculiar it is hard to develop some intuition
  24. P

    Can a Basis Be Proven with Divisibility?

    Homework Statement Hi everyone. I'm studying a problem and I need to prove that I have a basis. I tryed a proof and to achieve it I need to show that : if k divides a*b and also divides a2 +2*b2 Then k divides both a and b. Homework Equations I'm not sure what I'm asserting is...
  25. Mentz114

    Obtaining the Metric in a Boosted Observer Frame?

    I'm trying to get a metric in the frame of a boosted observer. The spacetime in question has coframe and frame basis vectors \begin{align*} \vec{\sigma}^0 = \frac{-1}{\sqrt{F}}dt\ \ \ \ & \vec{e}_0 = -\sqrt{F}\partial_t \\ \vec{\sigma}^1 = \sqrt{F}dz\ \ \ \ & \vec{e}_1 =...
  26. A

    Basis and dimension of the solution space

    Homework Statement Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations. x - 2y + z = 0 y - z + w = 0 x - y + w = 0 Homework Equations The Attempt at a Solution (a) [1 -2 1 0] => [1 0 -1 2] [0 1 -1 1] => [0 1...
  27. P

    Finding basis for kernal of linear map

    Homework Statement Let A = 1 3 2 2 1 1 0 -2 0 1 1 2 Viewing A as a linear map from M_(4x1) to M_(3x1) find a basis for the kernal of A and verify directly that these basis vectors are indeed linearly independant. The Attempt at a...
  28. J

    Angular momentum/Hamiltonian operators, magnetic field, basis states problem?

    Hi, Here's my problem, probably not that difficult in reality but I don't get how to approach it, and I've got an exam coming up soon... An atom with total angular momentum l=1 is prepared in an eigenstate of Lx, with an eigenvalue of \hbar. (Lx is the angular momentum operator for the...
  29. H

    Inner product as integral, orthonormal basis

    Homework Statement Define an inner product on P2 by <f,g> = integral from 0 to 1 of f(x)g(x)dx. find an orthonormal basis of P2 with respect to this inner product. Homework Equations So this is a practice problem and it gives me the answer I just don't understand where it came from...
  30. B

    Finding a basis and dimension of a subspace

    Homework Statement Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]} Let W=spanS. Find a basis for W. What is dim(W)? Homework Equations The Attempt at a Solution i know that a basis is composed of linearly independent sets. This particular problem's...
  31. H

    Orthonormal basis spanned by 2 matrices

    Homework Statement Let M1 = [1 1] and M2 = [-3 -2] ________[1 -1]_________[ 1 2] Consider the inner product <A,B> = trace(transpose(A)B) in the vector space R2x2 of 2x2 matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R2x2 spanned by the...
  32. N

    Find Basis for Ker L & Range L | L(x,y,z,w) = (x+y, z+w, x+z)

    L: R^4 => R^3 is defined by L(x,y,z,w) = (x+y, z+w, x+z) A) Find a basis for ker L We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). I then reduced it to row echelon form We now have the equations X-W=0 , Y+W=0, Z+W=0. There are infinitely many...
  33. C

    Are My Basis Calculations for R3 and R4 Subspaces Correct?

    Homework Statement Find the basis for the subspaces of R3 and R4 below. Homework Equations A) All vectors of the form (a,b,c), where a=0 B) All vectors of the form (a+c, a-b, b+c, -a+b) C) All vectors of the form (a,b,c), where a-b+5c=0 The Attempt at a Solution I honestly had...
  34. T

    Basis for the homogeneous system

    Homework Statement Find a basis for the solution space of the homogeneous systems of linear equations AX=0 Homework Equations Let A=1 2 3 4 5 6 6 6 5 4 3 3 1 2 3 4 5 6 and X= x y z...
  35. R

    Partial derivatives as basis vectors?

    Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...
  36. Z

    Linear transformation of an orthonormal basis

    Homework Statement Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for...
  37. 1

    Basis for the indicated subspace

    hi guys, i have no idea of how to do the following question, could u give some ideas? Q:determine whether or not the given set forms a basis for the indicated subspace {(1,-1,0),(0,1,-1)}for the subspace of R^3 consisting of all (x,y,z) such that x+y+z=0 how should i start? i know the...
  38. P

    Linear Algebra: Vector Basis (change of)

    Homework Statement Given one known orthonormal basis S in terms of the standard basis U, how would I express a third basis T in terms of U when I know its representation in S? For example, U consists of <1,0,0> <0,1,0> <0,0,1> And S consists of (for example) <0.36, 0.48, -0.8>...
  39. J

    Packing fraction for multi-atom basis

    Hi, I understand that the maximum packing fraction for a particular atomic structure can be calculated assuming the nearset neighbours are touching but my question is how can the maximum packing fraction be calculated for a basis containing two different types of atoms? Thanks, James
  40. G

    Basis of Orthogonal Complement

    Let S be the subspace of R^3 spanned by x=(1,-1,1)^T. Find a basis for the orthogonal complement of S. I don't even know where to start... I would appreciate your help!
  41. D

    Understanding Shankar's Principles of QM: Changing Basis of Operators

    Hi, I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?
  42. H

    Coordinates relative to a basis (linear algebra)

    Homework Statement The set B = {-4-x^2, -8+4x-2x^2, -14+12x-4x^2} is a basis for P2. Find the coordinates of p(x) = (-2 +0x -x^2) relative to this basis. Homework Equations n/a The Attempt at a Solution so the set would be in a matrix like this: |-4 0 -1|...
  43. P

    What is the Basis for an Extension Field Adjoined with an Element?

    I am having no luck understanding how to find the basis of a field adjoined with an element. For example Q(sqrt(2)+sqrt(3)) I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero. So, from that I know the...
  44. G

    Orthonormal basis for subsets of C^3

    We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3 a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai Thus a vector x in W_perp is x = (a,b,-ai) So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 +...
  45. L

    What is the matrix of T with respect to the basis (2,1),(1,-2)?

    Homework Statement Recall that the matrix for T: R^{2} \rightarrow R^{2} defined by rotation through an angle \theta with respect to the standard basis for R^{2} is \[A =\begin{array}{cc}cos \theta & -sin \theta \\sin \theta & cos\theta \\\end{array}\]\right] a) What is the matrix of T...
  46. R

    Finding the basis for a set of polynomials (linear algebra)

    Hi. Thanks for the help. Homework Statement Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0. Homework Equations P' is the first derivative, P'' is the second derivative. The Attempt at a Solution The general form of a polynomial in P3 is ax^3+bx^2+cx+d...
  47. B

    Basis for Vector Space: Understanding the Exceptional Case

    Homework Statement My notes has the following statement, but I seem to have forgotten to write down the conclusion of the statement before my professor erased it from the board. "Any vector space V there will be a basis except for 1 type of space: " Any ideas as to what that 1 type of...
  48. F

    Abstract Linear Algebra: Dual Basis

    Homework Statement Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0. Homework Equations N/A The Attempt at a Solution Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be...
  49. Dale

    Maxwell's Eqns: EM Basis Functions Explored

    Are there any sets of basis functions that are particularly useful for Maxwell's equations? I was thinking about Fourier just because it is the first basis I always think of, but I don't know that it would actually be a convenient basis. For example, I don't know that curl or divergence would...
  50. G

    Coordinate vector basis proving question

    Homework Statement Let S = {v1,v2,...,vn} be a basis for an n-dimensional vector space V. Show that {[v1]s,[v2]s,...[vn]s} is a basis for Rn. Here [v]s means the coordinate vector with respect to the basis S. Homework Equations [v]s is the coordinate vector with respect to the basis S...
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