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

    Finding Basis for S in LA: Solution

    Homework Statement Let S = [x y z w] \in R^4 , 2x-y+2z+w=0 and 3x-z-w=0 Find a basis for S. Homework Equations The Attempt at a Solution I started by putting the system into reduced row form: [2 -1 2 1] [3 0 -1 -1] [2 -1 2 1] [0 3 -8 -5] [6 0 -2 -2]...
  2. F

    Is pure mathematics the basis for all thought?

    I have been thinking much on the nature of pure mathematics. I believe this forum would make the best place to post over say the philosophy section, as i am more interested in the opinions of working mathematicians and physicists than philosophers. In my opinion pure mathematics is the core...
  3. fluidistic

    Cross product expressed in the cylindrical basis

    Homework Statement It's not a homework question but a doubt I have. Say I want to write \vec A \times \vec B in the basis of the cylindrical coordinates. I already know that the cross product is a determinant involving \hat i, \hat j and \hat k. And that it's worth in my case...
  4. estro

    Jordan Basis for Matrix A: Missing Something Trivial?

    In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?
  5. N

    What is the continuum of sexual preference and its role in evolution?

    Hello! I have been contemplating this question for a few days now and I am interested to know if anyone here has any input on the matter, and to critique my reasoning. There are other threads on this, none of which, from what I could find, made a distinction between homosexual behaviour and...
  6. B

    Matrix & Basis: Find D Matrix for V

    Consider a) f1=1, f2=sinx , f3=cosx b) f1=1, f2=ex , f3=e2x c)f1=e2x , f2=xe2x f3=x2e2x in each part B={f1,f2,f3} is a basis for a subspace V of the vector space. Find the matrix with respect to B of the differentiation operator D:V→V
  7. M

    How to extend a set to form a basis?

    i want to extend the set S={(1,1,0,0),(1,0,1,0)} to be a basis for R4. I know I am going to need 4 vectors, so i need to find 2 more that aren't linear combinations of the first 2. Is there a better way to approach this other than choose 2 at random and check linear independence/dependence...
  8. M

    Proving Invertible Matrix A Creates New Basis from {X1,X2,X3..Xn}

    does multiplying the invertible matrix A to the basis {X1,X2,X3..Xn} create a new basis; {AX1,AX2,Ax3..AXn}? where Xn are matrices I can prove that for eg if {v1,v2,v3} is a basis then {u1,u2,u3} is a basis where u1=v1 u2=v1+v2, u3=v1+v2+v3 I setup the equation c1(u1)+c2(u2)+c3(u3)=0 and...
  9. M

    Finding basis and dimension of W = {(a,b,c,0)} where abc are real numbers

    With normal vectors i usually check there is the correct number of vectors i.e 3 for R3 2 for R2 etc and then just check for linear independence but reducing the matrix that results from c1v1+c2v2+..cnvn=0 and determining of unique solution or infinite solutions. There are the right number of...
  10. F

    Reciprocal basis and orientation

    How to prove that two reciprocal basis are either both right ended or both left-handed? If (e_1,e_2,e_3) and (e^1,e^2,e^3) are two such basis, since the scalar triple products depend on orientation, it would be enough to show that VV'=1 (where V and V' are the volumes, taken with their sign, of...
  11. P

    What's the difference between a dictionary and a basis?

    I see the term dictionary used a lot and it sounds a lot like a basis for a vector space. But what is the different? Can we collection of vectors be both a basis and a dictionary? Thanks
  12. H

    Spin hamiltionian with respect to certain basis

    Homework Statement A particle with spin 1/2 and magnetic moment is in a magnetic field B=B_0(1,1,0). At time t=0 the particle has the spin 1/2 \hbar in the z direction. i) Write the hamiltonian with respect to the basis that is defined by the eigenvectors of \widehat{S}_z Homework...
  13. U

    Software to find basis of space

    Hi there, I have 23D subspace, defined by an equation (hyperplane) c1*x1 + ... + c24*x24 = 0; I wonder if there is an automated way to find basis of the subspace? I have access to Maple and Matlab. Thanks.
  14. F

    Water divining or witching What is the basis behind it?

    Water "divining" or "witching". What is the basis behind it? Before I start, let me say that I am hugely skeptical but we do have an issue where finding a good hitting water well might be tough so thought I might check...lol For those that are unaware, people are basically using pieces of...
  15. P

    Orthonormal basis => vanishing Riemann curvature tensor

    Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...
  16. M

    Proving The Hamming Metric: Open Subsets and Basis of X

    Homework Statement I'm stuck on how to start this. The Hammin metric is define: http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg and I'm asked to: http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360 a) prove...
  17. D

    Notation for basis of tangent space of manifold

    I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...
  18. L

    Dot product in non-orthogonal basis system

    Hi again, I don't want it to seem like I'm spamming topics here, but I was hoping I could get help with this dillema, too. So, let's say that, in affine 2-dimensional space, we have some two, non-orthogonal, independent vectors, and we also pick some point for an origin O. This clearly...
  19. E

    I am still confused how to prove that a set is a basis other than

    i am still confused how to prove that a set is a basis other than proving it linearly independent and system of generator that have to do with matrices? please help
  20. B

    How do I calculate the Basis for Im(T)?

    How do I calculate the Basis for Im(T)? I am having troubles finding an example that will best fir here. I know that the I=diagonal matrix with all of all of the i=j entries being 1. Beyond that I am rather confused and don't know where I need to start.
  21. M

    Basis of vector spaces over fields

    If you find the exact same basis for two vector spaces, then is it true that the vector spaces are equal to each other?
  22. Y

    Basis for the vector space of idempotents

    Homework Statement A matrix a is idempotent if a^2=a. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents 2. The attempt at a solution the vector space in question is dimension 4, so I need to find 4 idempotent matrices. but i don't want to find them...
  23. M

    Jordan Normal Form / Jordan basis

    Homework Statement Determine the Jordan Normal form and find some Jordan basis of the matrix 3 -3 1 A = 2 -2 1 2 -3 2Homework Equations dim(A) = rk(A) + dimKer(A The Attempt at a Solution My problem here is that my lecturer seems to be doing completely different things with...
  24. T

    Solving Linear Space Basis: Find q(t) & q'(t) Dimension

    Homework Statement Find a basis of the linear space and thus determine the dimension. q(t) : q'(1)=q(2) Where q is a subset of Q_2: all polynomials of degree less than or equal to 2. I had to change it from p to q so the forum didn't make it a smiley... The Attempt at a Solution...
  25. M

    Jordan Normal Form / Jordan basis

    Homework Statement Determine the Jordan Normal form and find some Jordan basis of the matrix 3 -3 1 A = 2 -2 1 2 -3 2 Homework Equations dim(A) = rk(A) + dimKer(A) The Attempt at a Solution My problem here is that my lecturer seems to be doing...
  26. C

    Linear transformation, basis of the image

    Homework Statement From Calculus we know that, for any polynomial function f : R -> R of degree <= n, the function I(f) : R -> R, s -> ∫0s f(u) du, is a polynomial function of degree <= n + 1. Show that the map I : Pn -> Pn+1; f -> I(f), is an injective linear transformation, determine...
  27. M

    Show that I have a basis, linear algebra

    Homework Statement (X,<,>) is a inner product space over R {ei}i in N is an orthonormal set in X Show that if every element u in X can be written as a linear combination u = \sum_{i=1}^\infty a_i e_i then {ei}i in N is a basis for X Homework Equations Let {ei}be a sequence of...
  28. Femme_physics

    Electricity basis resistors/power/current question

    Homework Statement A voltage of a car battery is 12V, the current requires for movement is 200A. When we turn the lights on the voltage drops to 11.1V. The power of each backlight is 55 watt, the power of each front light is 12 watt. A) Calculate the internal resistance of the battery...
  29. B

    Transforming Linear Transformation with Non-Standard Basis

    Homework Statement Suppose the matrix standard matrix for a linear trnaformation T: R^2 --->R^2 is[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Cbegin%7Bequation%7D%5Cbegin%7Bpmatrix%7D2%20%26%20-3%20%5C%5C%200%20%26%201%5Cend%7Bpmatrix%7D%5Cend%7Bequation%7D.gif Find the matrix T with...
  30. T

    Change of Basis Matrix for R2: B1 to B2

    Homework Statement B1 = {[1,2], [2,1]} is a basis for R2 B2 = {[1,-1], [3,2]} is a basis for R2 Find the change of basis matrix from B1 to B2 Homework Equations [B2 | B1] The Attempt at a Solution For some reason I can not solve this. I keep ending up with the matrix...
  31. tom.stoer

    Unique basis of relativistic field equations for arbitrary spin?

    Looking at Lagrangians and field equations for different spin all the derivations seem to lack a common basis; they appear to lack any deep relation. Is there a unique way to understand the different forms like Klein-Gordon, Dirac, Maxwell (Yang-Mills), etc. from a common basis which is valid...
  32. N

    Linear Algebra ~ Method to determine if a sequence of vectors is a basis

    Homework Statement A lot of my homework asks me to determine if a given matrix (sequence of vectors) is a basis or not. Homework Equations The Attempt at a Solution Can I just find the reduced echelon form of a given matrix and see if it is linear independent or linear...
  33. J

    Find a basis for the following subspaces

    Homework Statement Homework Equations In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system".[1] In more general...
  34. 7

    Basis + covariant derivative question

    My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold... Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or...
  35. S

    Extend Vector to Orthogonal Basis

    How do you extend a vector (let's use vector (1,2,3) for example) to an orthogonal basis for R^3?
  36. T

    Finding a basis for null(T) and range(T)

    Homework Statement Let Y:P3(R) onto P2(R) is defined by T(a0+a1z+a2z2+a3z3)=a1+a2z+a3z2. Find bases for null (T) and range (T). What are their dimensions? Homework Equations The Attempt at a Solution Well, I am assuming the range is the a1+a2z+a3z^2, so it has dimension of 3, and...
  37. E

    Density matrix and von Neumann entropy - why does basis matter?

    Density matrix and von Neumann entropy -- why does basis matter? I'm very confused by why I'm unable to correctly compute the von Neumann entropy S = - \mathrm{Tr}(\rho \log_2{\rho}) for the pure state | \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2 Now, clearly the simplest...
  38. L

    How do you find the basis for the spanning set which contains matrices?

    Please look at the link: http://gyazo.com/5f57caddf7dc76aa61d387f2915d29fe.png
  39. T

    Linear Algebra: Orthonormal Basis

    Homework Statement Find an orthonormal basis for the subspace of R^4 that is spanned by the vectors: (1,0,1,0), (1,1,1,0), (1,-1,0,1), (3,4,4,-1) The Attempt at a Solution When I try to use the Gram-Schmidt process, I am getting (before normalization): (1,0,1,0), (0,1,0,0), (1,0,-1,2)...
  40. B

    Usefulness of Basis for a Vector Space, General?

    "Usefulness" of Basis for a Vector Space, General? Hi, Everyone: I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension" a student asked me what was the use or purpose of a basis for a vector space V. All I could think of is that bases allow us to...
  41. H

    How to find the basis for a set of vectors

    Homework Statement The question states: Consider the subset S of R^4 given by: S={(2,3,-1,7), (1,0,1,3), (0,3,-3,1), (12,15,-3,29)} i) Decide whether the vectors in S form a linearly independent set. ii) Let V be the vector subspace of R^4 spanned by the vectors of S, i.e: V=span{...
  42. H

    Prove Linear Independence of {[w1]s, [w2]s,...,[wk]s} in V

    1) Let S be an ordered basis for n-dimensional vector space V. Show that if {w1, w2, ..., wk} is a linearly independent set of vectors in V, then {[w1]s, [w2]s,...,[wk]s} is a linearly independent set of vectors in R^n. What I got so far is w1 = a1V1 + a2V2 + ... + anVn so, [w1]s = [a1...
  43. K

    What is the minimal polynomial for Q(a1/2, b1/2)?

    1.Find the degree and basis for Q(3^1/2,7^1/2) over Q. 2.For any positive integers a, b, show that Q(a^1/2+b^1/2)=Q(a^1/2,b^1/2) Ideas: 1. Well I know if I looked at (3)^1/2 over Q Then (3)^1/2 has minimal polynomial x^2-3, so degree 2 over Q (7)^1/2 has minimal polynomial x^2-7 so...
  44. T

    Find a Basis for the Subspace of R4

    Homework Statement Find a basis for the subspace of R4 spanned by S. Homework Equations S: {(2,9,-2,53), (-3,2,3,-2), (8,-3,-8,17), (0,-3,0,15)} I've attempted this using a matrix and row reducing it. I'm just not sure if there's another simpler way, as I keep on getting incorrect...
  45. W

    Finding a basis of the image of a linear transformation

    Homework Statement Let Ψ: Mat2x2(R) -> Mat2x2(R) be defined as: [a,b;c,d] -> [a+b, a-c; a+c, b-c] Find a basis for the image of Ψ. Homework Equations None, AFAIK. The Attempt at a Solution I started by using the standard basis, B, for Mat2x2(R) to get [u]B [with u in Mat2x2(R)] as...
  46. mccoy1

    Find X4 to Make {X1, X2, X3, X4} Linearly Independent

    Homework Statement Hi fellows, If we are given 3 vectors (e.g X1, X2, X3) in R^4, how would we find X4 such that {X1, X2, X3, X4} is a linearly independent set? Homework Equations The Attempt at a Solution I tried something like this: aX1 + bX2 + cX3 + dX4 =0, but it didn't...
  47. T

    Proving the Basis Property for a Set in a Vector Space with a Nonzero Scalar

    Homework Statement Let c be a real scalar not equal to zero. Prove that if a set S ={v1, v2, ... , vn} is a basis for V, then set S1= {cv1, cv2, ... , cvn} is also a basis of V. Homework Equations A set is a basis if it spans a subspace and is not collinear. The Attempt at a...
  48. S

    Equation of a Surface relative to a basis

    Equation of a Surface relative to the standard basis is X1^1 + 2X2^2 + 3x^3 -4x1x2 -4x2x3 =0 Now the question ask to find the equation of above surface relative to the cordinate system with basis vectors F1= (2/3,2/3,1/3) F2= (1/3,-2/3,2/3) F3= (2/3,-1/3,-2/3) Now i found the transition...
  49. 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...
  50. 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...
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