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. T

    Linear Algebra - Bilinear Forms and Change of Basis

    Homework Statement Find the matrix of f relative to Alpha' and Beta'. Alpha' = [(1,0,0), (1,1,0), (2,-1,1)] Beta' = [(-13,9), (10,-7)] The question originally reads that f is a bilinear form. I've found a (correct according to answer key) matrix A that is 3 -4 4 -5 -1 2...
  2. T

    Basis for kernel of linear transform

    Hey guys! I am having a major brain problem today, with this problem. L is a linear transform that maps L:P4\rightarrowP4 As such that (a1t3+a2t2+a3t+a4 = (a1-a2)t3+(a3-a4)t. I am trying to find the basis for the kernel and range. I know that the standard basis for P4 is...
  3. C

    Subspace matrix, dimension and basis

    So an example was the matrix: A = \left(\begin{array}{cccc} a&a+b\\ b&0\\ \end{array} \right) is a subspace of M2x2. and is the linear combination a*\left(\begin{array}{cccc} 1&1\\ 0&0 \end{array} \right) + b*\left(\begin{array}{cccc} 0&1\\ 1&0 \end{array} \right) Meaning it has...
  4. fluidistic

    Expression a vector in different basis

    Homework Statement Consider the vector \vec A whose origin is \vec r. 1)Express the vector \vec A in a basis of Cartesian coordinates, cylindrical and spherical ones. 2)Repeat part 1) if the origin is \vec r + \vec r_0. Homework Equations None given. The Attempt at a Solution 1)In...
  5. U

    The Dependence of Norm on Basis in Vector Spaces

    Hello. My question is: does the norm on a space depend on the choice of basis for that space? Here's my line of reasoning: If the set of vectors V = \left\{ v_1,v_2\right\} is a basis for the 2-dimensional vector space X and x \in X, then let \left(x\right)_V = \left( c_1,c_2\right)...
  6. L

    Basis and Dimension of matrices

    If S is subspace of R6x6 consisting of all lower triangular matrices, what is the dimension of S? Does anyone know the properties about dimension of lower triangular matrices?
  7. D

    Finding a Basis for S: Polynomials in P3 with Specific Form

    My mind is shot. Let S be a subspace of P[SUB]3[SUB] consisting of all polynomials of the form ax2+bx+2a+3b. Find a basis for S. I am not sure where to start.
  8. T

    Basis of range of a matrix relative to some bases

    Homework Statement Let T be the linear transformation of R5 into R3 that has the matrix A = 1 3 2 0 -1 2 6 4 6 4 1 3 2 2 1 relative to the bases [(1,1,1,1,1), (1,1,1,1,0), (1,1,0,0,0), (1,0,0,0,0), (0,0,0,0,1)] of R5 and [(1,1,1), (0,1,0), (1,0,0)] of R3. Find a basis for the range...
  9. N

    Representations and change of basis

    Hi guys 1) We are looking at a Hamiltonian H. I make a rotation in Hilbert space by the transformation {\cal H} = \mathbf a^\dagger\mathsf H \mathbf a = \mathbf a^\dagger \mathsf U\mathsf U^\dagger\mathsf H \mathsf U\mathsf U^\dagger\mathbf a = \mathbf b^\dagger...
  10. Z

    Linear Algebra- Quadratic form and change of basis

    Homework Statement Suppose that for each v = (x1, x1, ... xn) in Rn, q(v) = XTAX for the given matrix A. For the given basis B of Rn, find the expression for q(v) in terms of the coordiantes yi of v relative to B. a) A =...
  11. T

    Bessel equation & Orthogonal Basis

    I remember some of my linear algebra from my studies but can't wrap my head around this one. Homework Statement Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...
  12. S

    Proof of Subspace and Basis Relationship in R^n - Homework Help"

    Homework Statement Prove or disprove this with counter example: Let U,V be subspaces of R^n and let B = {v1, v2,...,vr} be a basis of U. If B is a subset of V, then U is a subset of V. Homework Equations U and V are subspaces so 1. zero vector is contained in them 2. u1 + u2 is...
  13. M

    Dimension of Hom(K)(U,V) and Basis of the Vector Space

    Homework Statement Let U and V be vector spaces of dimensions of n and m over K and let Hom(subscriptK)(U,V) be the vector space over K of all linear maps from U to V. Find the dimension and describe a basis of Hom(subscriptK)(U,V). (You may find it helpful to use the correspondence with mxn...
  14. R

    Show that two vectors are a basis of a subspace

    Question: In R3, show that (1,-1,0) and (0,1,-1) are a basis for the subspace V={(x,y,z) \in R3: x+y+z=0} Attempt: By def of a basis, the vectors (1) must be linearly independent and (2) must span V. 1. For LI, show that if a(1,-1,0) + b(0,1,-1) = (0,0,0), then a=b=0...
  15. S

    Ordered basis and linear independence

    Homework Statement I have a set of Vector v_1,v_2,v_3,v_4 in \mathbb{R}^4 and need to show that E = v_1,v_2,v_3,v_4 is an ordered basis for \mathbb{R}^4 The Attempt at a Solution I know that for this being the case v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 where v \in...
  16. V

    Basis for Plane Perpendicular to a Line

    Homework Statement Let L be the line in R^3 spanned by v1=(1,1,1) Find a basis (v2,v3) for the plane perpendicular to L, and verify that B=(v1,v2,v3) is a basis for R^3. Homework Equations The Attempt at a Solution I know that if two vectors are perpendicular or orthogonal that...
  17. G

    Another linear algebra problem, basis and linear transformations.

    Homework Statement The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2. Homework Equations - The Attempt at a Solution I have no...
  18. M

    Basis and Dimension of Subspace V

    Homework Statement V = the set of all symetrical nXn matrices, A=(ajk) such that ajk=akj for all j,k=1,...,n Determine the base and dimensions for V The Attempt at a Solution I set my matrix up as [a11 a12] [a21 a22] So a21 and a12 are equal to each other? I assume the...
  19. M

    Basis for 2x2 Matrix: Understand the Concepts

    Homework Statement What is a basis for the space of 2 x 2 matrices. The Attempt at a Solution I don't understand how to this at all. Is the 2x2 identity matrix abasis for 2x2 matrices? Because it's linearly independent and spanning the space? Can anyone explain?
  20. C

    Understanding Subspace Basis and Counterexample

    My book made the following claim... but I don't understand why it's true: If v_1, v_2, v_3, v_4 is a basis for the vector space \mathbb{R}^4, and if W is a subspace, then there exists a W which has a basis which is not some subset of the v's. The book provided a proof by counterexample...
  21. Z

    Linear Algebra - Change of basis matrices and RREF question what in the world?

    Linear Algebra - Change of basis matrices and RREF question what in the world?? Homework Statement Suppose the linear transformation T: P3 -> P2, over R has the matrix A = \begin{bmatrix}1&2&0&0\\0&1&2&1\\1&1&1&1 \end{bmatrix}...
  22. S

    Basis for ROW(A), COL(A) and NUL of a square matrix

    deleted No one answered
  23. M

    Understanding the Basis of a Zero Vector Space

    Homework Statement erm, I just want to know, what is the basis for a zero vector space? Homework Equations The Attempt at a Solution is it the zero vector itself? but if that's the case, then the constant alpha could be anything other than zero, which means the zero vector is not...
  24. Z

    Linear Algebra - Change of basis question

    Homework Statement Let A = E4 in R4 (standard basis) and B = {x^2, x, 1} in P2 over R. If T is the linear transformation that is represented by \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix} relative to A and B, find...
  25. Z

    Linear algebra - change of basis matrix

    Homework Statement Let A = {(1, 1), (2,0)} and B = {(0, 2), (2, 1)} in R2. a) Find [u]A (u with respect to A) if [u]B = [3, -2]. Homework Equations The Attempt at a Solution I tried to find [I]AB (transition matrix from B to A), then apply to [u]B, but couldn't represent (2, 1)...
  26. T

    What is the General Solution for Finding Orthonormal Bases in C^2?

    I'm teaching myself quantum mechanics and am learning about bra-ket notation. There is a particular operator used, the ket-bra (e.g. |X><X|). To understand it, I'm trying to come up with an orthonormal basis for C^2 as a simple case (i.e., the 2-dimensional vector space over the field of complex...
  27. M

    What are the subspaces of a matrix and how do bases relate to them?

    Hi, I have an assignment due and I have done most of the questions there are just a couple things I have left, if someone can help that would be amazing :) 1. In this problem we suppose that F is a field, A is an m by n matrix over F and that W is a subspace of Fm. (a) Show that U =...
  28. G

    Showing functions form a basis

    Homework Statement Show that the functions po(t)=1, p1(t)=t, p2(t)=1/2(3t2-1), and p3(t)=(3/2)*[(5/3)t3-t) also form a basis for the vector space P3(R) ... "R" meaning all real numbers Homework Equations I know these polynomials are the first four Legendre polynomials The Attempt...
  29. J

    Finding basis for null(A) and null(AP)

    Homework Statement A is a mxn matrix, and P is an invertible nxn matrix. So I want to prove that the bases of null(A) and null(AP) have the same number of elements. Homework Equations The Attempt at a Solution I was going to start off by assuming that {X1, X2, ... Xm} is a...
  30. F

    What is the basis and dimension of a subspace given by (1,-2,3,-1), (1,1,-2,3)?

    say we are given a subspace like this: Being W the subspace of R generated by (1,-2,3,-1), (1,1,-2,3) determine a basis and the dimension of the subspace. Won't the vectors given work as a basis, as long as they are linearly independent? If so, all we have to do is check for dependance, and if...
  31. J

    Showing that something is a basis of an independent set

    Homework Statement Among all independent vector sets in a vector space U, let M = {v1, v2, ... vp} be an independent set. p is as large as it can get. Show that U is a basis of M. Homework Equations The Attempt at a Solution If U is a basis of M then U is an independent set (we...
  32. M

    Solving for Vector Space V: Find Dimension & Basis

    Homework Statement Find the dimnesion and a basis of vector space V Homework Equations V is the set of all vectors (a,b,c) in R^3 with a+2b-4c=0 The Attempt at a Solution (4c-2b,b,c) = b(-2,1,0) + c(4,0,1) so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right? how do I...
  33. C

    Dimension and basis for subspace determined by given vectors

    Homework Statement Assume that e_1 ,..., e_n is a basis for the vector space V. Let W be the linear subspace determined (formed?) by the vectors e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1}. Determine the dimension of W, and a basis for W. Homework Equations The...
  34. CFDFEAGURU

    Hartle Gravity - Simple basis vector question

    Helo all, I have a very simple question about basis Four-Vectors and Components. In Hartle's book, Gravity, he uses the following equation to show the components of the 4-vector, a a =a^t{}e(sub t) + a^x{}e(sub x) + a^y{}e(sub y) + a^z{}e(sub z) Sorry for the half LaTex half something...
  35. E

    Could Ammonia be the Basis of Life?

    I'm guessing this is the right forum to post in. Ammonia shares many properties with water. It is polar, it is amphoteric, it reacts with itself to form its acid and base conjugates NH4+ and NH2-. Just as water is our basis of life, could another species use ammonia like we use water...
  36. S

    Time as a Basis Vector in Quantum Mechanics

    I was explaining basis vectors to my brother, I said that in quantum mechanics that when you have a number of dimensions, each dimension being an eigenket in vector space, that every dimension is independent of all the other basis vectors. It is however interesting to think that if this is the...
  37. K

    Basis vectors under a Lorentz transformation

    Hello, I am new to the forums and I hope this fundamental topic has not been previously treated, as these forums don't seem to have a search function. I am studying general relativity using S. Carroll's book (Geometry and Spacetime) and I am having a fundamental problem with basis vectors under...
  38. S

    Basis and Subspace Help: Exploring the Relationship between Vectors e and d

    Suppose I have 3 vectors e1, e2, e3 that spans the subspace E, another 3 vectors d1, d2, d3 that spans the subspace D. If I also know that e1’d1 = 0, e2’d2 = 0, e3’d3 = 0, are there any conclusions I can make in terms of E and D? like row(E) = null(D)?
  39. T

    What is the so called orthogonal operator basis ?

    What is the so called "orthogonal operator basis"? What is the so called "orthogonal operator basis"?
  40. G

    Basis of mathematical deduction

    I want to write a program that can do algebraic transformations and mathematical deduction for me. It's not meant to do anything by itself, but rather check the transformations that I do myself for validity. The set of rule I will specify in advance. I want to capture all/most of the...
  41. nomadreid

    Basis for conservation of mass-energy?

    As far as I can make out, the hallowed principle of conservation of mass-energy (modulo quantum fluctuations) lies on four principles: (1) from the assumption that every effect has a cause (again, modulo the leeway given by Heisenberg), so if there is no mechanism for creation or destruction of...
  42. P

    Basis Transformation for Wave Function

    Homework Statement It's not a homework problem. I'm reading my textbook (Sakurai's Modern QM), and I'm not sure about a step (eq 3.6.6 through 3.6.8). Here it is: We start with a wave function that's been rotated: \langle x' + y' \delta \phi, y' - x' \delta \phi, z' | \alpha \rangle Now...
  43. S

    Solve Linear Algebra Exam: Find a Basis of U & Orthogonal Complement

    I have my linear algebra exam coming up but I missed the class on bases. Can anyone show me how this is solved? 2. Consider the subspace U of R4 defined by U = span{(−1, 1, 0, 2), (1, 0, 0, 1), (2,−1, 1,−1), (0, 1, 0, 3)} • Find a basis of U. • Find a basis of the orthogonal complement U.
  44. N

    Is there a biological basis for helping people?

    A few mornings ago, I helped a young woman with her car (dead battery), which made me late for work. Which brought some negative consequences to me. Yet I would do the same thing again just to help. With no benefit to me, but with negative consequences, I felt good for helping someone in...
  45. H

    The preferred basis problem - help

    Could someone please tell me what the preferred basis problem is with regards to the Everett/many worlds interpretation? As I understand it, it refers to the basis which is needed to make macroscopic objects determinate in all worlds. But what does this mean? Does basis refer to an observable...
  46. C

    Kernel, Range, Basis (linear algebra)

    Hey all! I am working on this and got confused. Any help at all would be much appreciated! Determine the kernel and range of the transformation T and find a basis for each: T(x,y,z)=(x,y,z) from R3 to R2. I have found the kernel to be the set {(r, -r, 0)}. Range is R2. I"m not sure how...
  47. F

    Finding a Basis Subset in (a, b, c, d) for S in R4 | Homework Solution

    Homework Statement Let S be the form of (a, b,c ,d )in R4, given a not equal to 0. Find the basis that is subset of S.Homework Equations The Attempt at a Solution I got a(1,0,0,0), b(0,1,0,0), c(0,0,1,0), d(0,0,0,1) as basis. a not = 0 But i wasn't sure what the significances of a not = to 0...
  48. D

    Expressing Vector w/o Basis: Dirac Bra-Ket Notation

    Inspired by the Dirac bra-ket notation I came to think that an ordinary Euclidean vector must be expressible without reference to a basis. But if I specify the length and angle of a vector, I have to refer this angle to some particular direction. Isn't this the same as choosing a basis? Edit...
  49. N

    Vector Space Basis: Standard or Odd?

    In short: does every vector space have a "standard" basis in the sense as it is usually defined i.e. the set {(0,1),(1,0)} for R2? And another example is the standard basis for P3 which is the set {1,t,t2}. But for more abstract or odd vector spaces such as the space of linear transformations...
  50. J

    Linear Algebra Finding Basis for space.

    Homework Statement Says, The set W = {(x,y,z,w) : x+z=0, 2y+w=0} is a subspace of R^4. Find a basis for W, and state the dimension. The Attempt at a Solution What I did: W= {(-z,-w/2,z,w): z,w are in R} = {z(-1,0,1,0) + w(0,-1/2, 0, 1)} = span {(-1,0,1,0), (0,-1/2,0,1)} (-1,0,1,0)...
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