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. Sai-

    Basis for a Plane that is given.

    Homework Statement Consider the plane 3x1-x2+2x3 = 0 in R3. Find a basis for this plane. Hint: It's not hard to find vectors in this plane. Homework Equations Plane: 3x1-x2+2x3 = 0 in R3. The Attempt at a Solution Let, A = \left[3,\right.\left.-1,\right.\left.2\right] \rightarrow...
  2. N

    Basis for Subspace: Find & Check LI

    Homework Statement Find the basis for the subspace 4x+y-3z The Attempt at a Solution I found that the basis is {[1;-4;0],[0;3;1]}. How do I know if it is linearly independent? I know that the mathematical definition of what LI is but how can it be applied to show in this case?
  3. N

    Linear transformation and change of basis

    Homework Statement Let B = {(1, -2),(2, -3)} and S be the standard basis of R2 and [-8,-4;9,4] be a linear transformation expressed in terms of the standard basis? The Attempt at a Solution 1) What is the change of basis matrix PSB ? 1,2 -2,-3 2)What is the change of...
  4. C

    Constructing a chart with coord. basis equal to given basis at one pt.

    Suppose we have a manifold ##M## and at ##p \in M## we have a basis for the tangent space of vectors ##X_i##. Since ##M## is a manifold, there exists a local chart ##(U,\phi)## about ##p##. Now the question is, given such a chart , how can we construct a new chart in a such that ##X_i = \left...
  5. W

    Adjointness and Basis Representation

    Hi, Let V be a fin. dim. vector space over Reals or Complexes and let L: V-->V be a linear operator. I am just curious about how to use a choice of basis for general V, to decide whether L is self-adjoint. The issue, specifically, is that the relation ## L= L^T ## ( abusing notation ; here L...
  6. B

    Riemann Manifold: Choosing a Basis & Lie Algebra

    On the spacetime manifold in general relativity, one chooses a basis at a point and express it by the partial derivatives with respect to the four coordinates in the coordinate system. And then the basis vectors in the dual space will be the differentials of the coordinates. Why do one do that...
  7. V

    Determine a basis for the image of F

    Hello I have a question about part (b) Now I know that F(f)(x) = [##\frac{1}{2}a+\frac{5}{6}b+\frac{17}{12}c]+[-a-2b-4c]x+[b+\frac{7}{2}c]x^{2}+[\frac{-2}{3}b-\frac{7}{3}c]x^{3}## so Kernel of F : F(f)(x) =0 => ##\frac{1}{2}a+\frac{5}{6}b+\frac{17}{12}c=0## ##-a-2b-4c=0##...
  8. G

    Bound States, Negative Potential, Alternate Basis, Matrix Mechanics

    Homework Statement Given the potential V(x) = - 1/ sqrt(1+x^2) Consider this in a 50x50 matrix representation of the hamiltonian in the basis of a one dimensional harmonic oscillator. Determine the eigenvalues and eigenvecotrs, the optimal parameter for the basis, and cop ate the...
  9. G

    Bound state negative potentials into harmonic oscillator basis

    Hello readers, Given the potential V(x) = - 1/ sqrt(1+x^2) I have found numerically 12 negative energy solutions Now I want to try to solve for these using matrix mechanics I know the matrix form of the harmonic oscillator operators X_ho, P_ho. I believe I need to perform the...
  10. C

    Rotation and translation of basis to remove cross terms

    So in our notes we are given a general quadratic equation in three dimensions of the form: Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0 And then they say, by some rotation we can change this to the standard form: Ax^2 + By^2 + Cz^2 + J = 0 The lecturer said don't worry...
  11. Math Amateur

    MHB Basis of a vector space - apparently simple problem

    I am revising vector spaces and have got stuck on a problem that looks simple ... but ... no progress ... Can anyone help me get started on the following problem .. Determine the basis of the following subset of \mathbb{R}^3 : ... the plane 3x - 2y + 5z = 0 From memory (I studied vector...
  12. J

    If a irrational number be the basis of count

    In the comum sense, the number 10 is the base of the decimal system and is the more intuitive basis for make counts (certainly because the human being have 10 fingers). But, in the math, the number 10 is an horrible basis when compared with the constant e. You already thought if an irrational...
  13. P

    Growing Basis in R: Exploring Primes & Integers

    Hi i was just wondering if there is any concept/theory/idea (or anything really) that relates to a growing basis (primarily in R) . What i mean by a growing basis is the following; say you start off with one element in your basis and you encounter a number/vector that cannot be built by the...
  14. V

    Component is diagonal in the basis

    Hi, I am analyzing the paper for my thesis and have Equation IF=-1*(1+cos(δ))*cot(θ)*σ_2-sin(δ)*cot(θ)*σ_1 where σ_1={{0,1},{1,0}} and σ_2={{0,-i},{i,0}} are the Pauli matrices The component σ_1 is diagonal in the (1/√2, +/- 1/√2) linear polarization basis and the component σ_2 is...
  15. nomadreid

    Confusion about basis vectors and matrix tensor

    In "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, I read that the covariant metric tensor gij=ei°ei (I'm leaving out the → s above the e's) where ei and ei are coordinate basis vectors and ° denotes the inner product, and similarly for the contravariant metric tensor using dual...
  16. M

    Find a Basis B for the subspace

    Homework Statement Let V be the subspace of R3 defined by V={(x,y,z)l2x-3y+6z=0} Find a basis B for the subspace. Homework Equations The Attempt at a Solution First I broke apart the equation such that: [[x,y,z]] = [[3/2s-3t, s, t]] = s [[3/2, 1 ,0]] +t[[-3, 0, 1]]...
  17. C

    Can any vector be in orthonormal basis?

    okay so I'm having some conceptual difficulty given some vector space V (assume finite dimension if needed) which has some orthonormal basis i'm given a vector x in V (assume magnitude 1 so it is normalized) now my question is: can x belong to some orthonormal basis of v...
  18. D

    Unitary matrix and preservation of vector norm in arbitrary basis

    Hi PF people! I am not sure my question can elegantly fit in the template, but I 'll try. Homework Statement I am self-studying the 8th chapter of "Mathematical Methods for Physics and Engineering", 3rd edition by Riley, Hobson, Bence. In the section about unitary matrices, it is stated that...
  19. A

    Matriz Lz, m=2, but in x,y,z basis

    I have read the follow bock...
  20. W

    Linear Algebra: Kernel, Basis, Dimensions, injection, surjections

    Homework Statement The Attempt at a Solution Can someone please check my work?
  21. W

    Basis of the range of a Linear Transformation

    Mod note: fixed an exponent (% --> 5) on the transformation definition. Homework Statement A is a (4x5)-matrix over R, and L_A:R^5 --> R^4 is a linear transformation defined by L_a(x)=Ax. Find the basis for the range of L_A. Homework Equations The Attempt at a Solution ##A =...
  22. N

    Find a basis for the null space of the transpose operator

    Homework Statement Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##. Homework Equations Let ##T:V\to W## be a linear...
  23. 9

    How to Express a Vector in Terms of Basis Vectors?

    Homework Statement Given the basis vector: e1 = 1 0 0 0 e2 = 0 1 0 0 e3 = 0 0 1 0 e4 = 0 0 0 1 Express the following vector in terms of the basis: y = 3 1 2 5 Homework Equations e1 = 1 0 0 0 e2 = 0 1 0 0 e3 = 0 0 1 0 e4 = 0 0 0 1 y = 3 1 2 5 The Attempt at a...
  24. N

    Systematic way of extending a set to a basis

    Homework Statement I want to extend the below U set of vectors to R4. u1 = (0, 0, 0, -4), u2 = (0, 0, -4, 3), u3 = (3, 2, 3, -2). The Attempt at a Solution For a set of vectors to form a basis for Rn, the vectors must be LI and spans Rn(has n vectors) u1, u2 and u3 are...
  25. M

    Linear function standard basis.

    Does anyone know what L is? I'm trying to see if I could find videos on it on YouTube. On the first question this is what I think- [a;b] is a vector by the way: 1) [2;1]c1+[7;4]c2=[1;0] [2;1]c1+[7;4]c2=[0;1] I could have also combined those two by having the linear combination equal to a size...
  26. T

    Linear Algebra - Basis of column space

    Homework Statement Let A be the matrix A = 1 −3 −1 2 0 1 −4 1 1 −4 5 1 2 −5 −6 5 (a) Find basis of the column space. Find the coordinates of the dependent columns relative to this basis. (b) What is the rank of A? (c) Use the calculations in part (a) to...
  27. E

    MHB Finding Basis for Intersection of Subspaces of $\Bbb R^n$

    I am looking for a method of finding a basis of the union and intersection of two subspaces of $\Bbb R^n$. My question is primarily about the intersection. Suppose that the basis of $L_1$ is $\mathcal{A}=(a_1,\dots,a_k)$ and the basis of $L_2$ is $\mathcal{B}=(b_1,\dots,b_l)$. Then $v\in L_1\cap...
  28. G

    What Are the Key Properties and Questions About Free Modules?

    We define basis as: Let M be a module over a ring R with unity and let S be a subset of M. Then S is called a basis of M if 1. M is generated by S 2. S is linearly independent set. Also we define free module as An R module M is called a free module if there exists a subset S of M s.t.S...
  29. N

    Is my Solution for [T]B Correct?

    Homework Statement Let A = [1 0 4 2 ] Let B be the eigenbasis {[1,4], [0,1]}. --Find [T]B where T(x)=A(x). The Attempt at a Solution Would [T]B = {[1,-1], [0,2]}? We are trying to find [T]B, the matrix representation of T with respect to B. So would my answer...
  30. E

    Why Can't Polar Basis Vectors Be Defined as Unit Vectors?

    I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain: We have our usual polar coordinates relation to Cartesian: x = r cosθ ; y = r sinθ if I define \hat{e_{r}}, \hat{e_{\vartheta}} as the polar basis vectors, then they should be...
  31. V

    Kernel, Basis, Rank: Hints & Answers

    Please see attached question In my opinion this question is conceptional and abstract.. For part a and b, I think dim(Ker(D)) = 1 and Rank(D) = n but I do not know how to explain them For part c What I can think of is if we differentiate f(x) by n+1 times then we will get 0 Can...
  32. I

    Guidance: Convex hull, null space and convex basis etc

    Hi friends! I am getting started with a research paper that discusses the closure properties of a robotic grasp. There are of lot of mathematical terms that confuse me like 'convex hull' , convex basis, convex combination of vectors, a free subset, nullspace etc. I might have studied some of...
  33. T

    Find the eigenstates of a basis in terms of those of another basis?

    Homework Statement This isn't exactly a homework question, but I figured this would be the best subforum for this sort of thing. For the sake of a concrete example, let's just say my question is: Express the position operator's eigenstates in terms of the number operator's eigenstates...
  34. M

    Transition matrix -> change of basis.

    Homework Statement B = {b1, b2, b3} and C = {c1, c2, c3} are two basis's for R3 where the connection between the basis vectors are given by b1 = -c1 + 4c2, b2 = -c1 + c2 + c3, b3 = c2 - 2c3 a) decide the transformation matric from basis B to basis C. A vector x is given in...
  35. F

    Calculating Basis of Tangent Plane

    I've looked at this topic for a while and I have yet to come to any sort of conclusive answer when it comes to calculating the basis of a surface's tangent vector. Do you have a concrete method or know where I can find one for doing this? Thank you
  36. T

    S_z matrix in s_y basis: three methods and three results

    Hello guys. when i write s_z matrix in s_y basis with unitary transformation i get s_y but in second method when i rotate s_z about x-axis ( - pi/2 ) to get s_z in s_y basis , i get -s_y ! and if we consider cyclic permutation, s_z becomes s_x. i am totally confused! three methods and three...
  37. C

    Preferred Basis and Superposition

    Hi Guys, I have a question about observable's and superposition's that I haven't been able to find a definitive answer to (purely for the fact that it doesn't seem to be addressed), and would greatly appreciate an answer. When in a superposition state of an observable, you are also in a...
  38. M

    Y^2 - x^2 in the [itex]\mid n\ell m \rangle[/itex] basis - tensor Op.

    x^2 - y^2 in the \mid n\ell m \rangle basis - tensor op. Homework Statement I must determine the matrix elements of x^2 - y^2 in the \mid n\ell m \rangle basis. "...use the fact that x^2 - y^2 is a sum of spherical components of a rank two tensor, together with the explicit form of the...
  39. carllacan

    Derivative of a parametrized vector on a nonfixed basis

    Homework Statement Find the parameter derivative of the vector function v(u, v) in, say, polar coordinates, i.e, this: http://en.wikipedia.org/wiki/Vector-valued_function#Derivative_of_a_vector_function_with_nonfixed_bases but deriving with respect to the parameter u or v instead of the...
  40. T

    Represent |+x> and |-x> in the Sy basis

    Homework Statement Determine the column vectors representing |+x> and |-x> using the states |+y> and |-y> as a basis. Homework Equations N/A The Attempt at a Solution I know that if |+y> and |-y> are used as a basis, then they are the column vectors (1,0) and (0,1) respectively. I...
  41. N

    Finding a Basis for P2 Subspace with p'(5)=0

    Homework Statement The problem asks, find a basis for the P2 subspace that consists of polynomials, p(x) such that p'(5)=0. The Attempt at a Solution I know that a set of vectors is a basis if it's linear independent and spans the vector space. So I let p(x) = ax2 + bx +c ...
  42. J

    Understanding the Basis of a Bivector: Exploring e_{11} and Its Significance

    What is the basis of a bivector? For example (see the attachment and http://en.wikipedia.org/wiki/Bivector#Axial_vectors first): e_{11}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{bmatrix} or e_{11}=\begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix} or ##e_{11}## is equal to what...
  43. Petrus

    MHB Extend to an orthonormal basis for R^3

    Hello MHB, (I Hope the picture is read able) this is a exemple on My book ( i am supposed to find a singular value decomposition) well My question is in the book when they use gram-Schmidt to extand they use (u_1,u_2,e_3) but I would use (u_1,u_2,e_1) cause it is orthogonal against u_2 which...
  44. H

    How to select the good basis for the special Hamiltonian?

    How to select the good basis for the special Hamiltonian?? For the Hamiltonian H=\frac{P^2}{2\mu} -\frac{Ze^2}{r}+ \frac{\alpha}{r^3} L.S (which we can use L.S=\frac{1}{2} (J^2-L^2-S^2)in the third term) how to realize that the third term,\frac{\alpha}{r^3} L.S, commutes with sum of the...
  45. R

    Exam in 2 days Change of basis BRA and KET

    ive been revising all holidays, unfortunately I've just realized I've been finding eigenvalues using the ensemble when i may have to change basis for the exam. looks at homework questions, workshop questions... nothing! anyway an example problem: rewrite |PSI> = a|0> + b|1> in the...
  46. Y

    MHB Basis, dimension and vector spaces

    Hello all, I have these two sets (I couldn't use the notation {} in latex, don't know how). V is the set of matrices spanned by these 3 matrices written below. W is a set of 2x3 matrices applying the rule a+e=c+f \[V=span(\begin{pmatrix} 1 &1 &1 \\ 1 &3 &7 \end{pmatrix},\begin{pmatrix} 0 &0...
  47. T

    How to Determine if a Particle Has Returned to the Origin?

    Homework Statement A point begins at rest at x = 0 and accelerates at 1.09 m/s^2 to the right for 10 s. It then continues at constant velocity of 10.9 m/s for 8 more seconds. In the third phase of its motion, it decelerates at 5 m/s^2 and is observed to be passing again through the origin when...
  48. D

    Vector components and its coordinate description in a given basis

    Given a basis \mathfrak{B}=\lbrace\mathbf{e}_{i}\rbrace it is possible to represent a vector \mathbf{v} as a column vector \left[\mathbf{v}\right]_{\mathfrak{B}}= \left(\begin{matrix}v^{1} \\ v^{2} \\ \vdots \\ v^{n}\end{matrix}\right) where the v_{i} are the components of \mathbf{v} relative...
  49. V

    Understanding Basis Choices in Quantum Mechanics

    Homework Statement Alright, so this is not exactly a guided homework question. It is a rather intricate problem consisting of many steps, but one of these steps comes down to working out the hyperfine interaction between two spin 1 particles, from a hamiltonian point of view. From what I have...
  50. J

    Do These Vectors Form a Basis for the Vector Space?

    Homework Statement Let v_1,...,v_k be vectors in a vector space V. If v_1,...,v_k span V and after removing any of the vectors the remaining k-1 vectors do not span V then v_1,...,v_k is a basis of V? Homework Equations The Attempt at a Solution If v_1,...,v_k span V but...
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