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).

View More On Wikipedia.org
Recent contents Top users
  1. B

    Multiplying matrix units and standard basis vectors

    Hello all, I don't have a question on homework specifically, but I need clarification on something I'm reading in the textbook. I will be starting an abstract algebra class in the spring and it's been quite a few years since I've had linear algebra, so I'll be reviewing that material before the...
  2. PhizKid

    How do you find the coordinates of a polynomial in terms of an orthogonal basis?

    Homework Statement Given ##S = \{1, x, x^2\}##, find the coordinates of ##x^2 + x + 1## with respect to the orthogonal set of S.Homework Equations Inner product on polynomial space: ##<f,g> = \int_{0}^{1} fg \textrm{ } dx## The Attempt at a Solution I used Gram-Schmidt to make ##S## orthogonal...
  3. Sudharaka

    MHB Finding Basis of the Quotient Space

    Hi everyone, :) This seems like a pretty simple question, but up to now I haven't found a method to solve it. Hope you can provide me a hint. :) Problem: Let \(V\) be a space with basis \(B=\{b_1,\,b_2,\,b_3,\,b_4,\,b_5\},\,U\) the subspace spanned by \(u_1=b_1+b_2+b_3+b_4+b_5\)...
  4. T

    MHB Existence of a Basis of a Vector Space

    Let n be a positive integer, and for each $j = 1,..., n$ define the polynomial $f_j(x)$ by f_j(x) = $\prod_{i=1,i \ne j}^n(x-a_i)$ The factor $x−a_j$ is omitted, so $f_j$ has degree n-1 a) Prove that the set $f_1(x),...,f_n(x)$ is a basis of the vector space of all polynomials of degree ≤ n -...
  5. N

    Find a basis for the subset

    Homework Statement Find a basis for the subset S = {(1, 2, 1), (2, 1, 3), (1, -4, 3)} Homework Equations The Attempt at a Solution I'm not absolutely sure I'm doing this correctly but here is my attempt: First, I put the vectors in S in the rows of a matrix (using multiple...
  6. Sudharaka

    MHB Bivector and Existence of Basis

    Hi everyone, :) I am trying to find an approach to solve this but yet could not find a meaningful one. Hope you can give me a hint to solve this problem. Problem: Prove that for any bivector \(\epsilon\in\wedge^2(V)\) there is a basis \(\{e_1,\,\cdots,\,e_n\}\) of \( V \) such that...
  7. S

    MHB Finding a Basis for a Bivector in $\Lambda^2 (V)$

    Hello everyone Can anyone help me to solve the following problem Prove that for any bivector $v\in \Lambda^2 (V)$ there is a basis $\{e_1,e_2,...,e_n\}$ of $V$ such that $v=e_1\Lambda e_2 +e_3\Lambda e_4 +...+e_{k-1}\Lambda e_k$ I did it in this way, since the out product $e_i\Lambda e_j$...
  8. V

    Is the empty set always part of the basis of a topology?

    The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...
  9. S

    Show aβ is a Basis for ℝ over Q

    Let β be a basis for ℝ over Q (the set of all rational numbers) and let a\inℝ, a≠1. Show that aβ={ay|y\inβ} is a basis for ℝ over Q for all a≠0. So I need to show (1) Linear independence, and (2) spanning. I am a little confused, especially because the dimension for the vector space is...
  10. Sudharaka

    MHB Canonical Basis and Standard Basis

    Hi everyone, :) I have a little trouble understanding what Canonical basis means in the following question. I thought that Canonical basis is just another word for the Standard basis. Hope you people could clarify the difference between these two in the given context. :) Question: Find the...
  11. Sudharaka

    MHB Transforming a Linear Transformation Matrix to an Orthonormal Basis

    Hi everyone, :) Here's a question with my answer. It's pretty simple but I just want to check whether everything is perfect. Thanks in advance. :) Question: Let \(f:\,\mathbb{C}^2\rightarrow\mathbb{C}^2\) be a linear transformation, \(B=\{(1,0),\, (0,1)\}\) the standard basis of...
  12. C

    What is a Basis of a Vector Space and How to Find Another Basis?

    Homework Statement There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4. I need to find another basis B' for R4[z] such that no scalar multiple of an element in B appears as a basis vector in B' and also prove that...
  13. C

    How to Find the Change of Basis Matrix for Bases B and C?

    Edit complete, but it doesn't seem as though I can change the title. The latex arrows next to the 'P' aren't showing up for me but they're supposed to be left arrows Homework Statement Let B and C be bases of R^2. Find the change of basis matrices P_{B \leftarrow C} and P_{C\leftarrow B}...
  14. M

    Einstein's Basis for Equivalence in his Field Equations

    The following is a question regarding the derivation of Einstein's field equations. Background In deriving his equations, it is my understanding that Einstein equated the Einstein Tensor Gμv and the Cosmological Constant*Metric Tensor with the Stress Energy Momentum Tensor Tμv term simply...
  15. A

    Hilbert space, orthonormal basis

    My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously...
  16. H

    Coordinate and dual basis vectors and metric tensor

    I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...
  17. H

    Using Fourier Sine basis to write x(L-x) [0,L]

    Homework Statement A function F(x) = x(L-x) between zero and L. Use the basis of the preceding problem to write this vector in terms of its components: F(x)= \sum_{n=1}^{\infty}\alpha _{n}\vec{e_{n}} If you take the result of using this basis and write the resulting function outside the...
  18. S

    Understanding Completeness of Fourier Basis

    So the other day in class my teacher gave a proof for the completeness of \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} in L^2([-\pi,\pi]) . And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel K_n(x) = \frac{1}{2\pi(n+1)}...
  19. Sudharaka

    MHB Understanding Matrix Units: A Guide for Linear Transformations in M2(Re)

    Hi everyone, :) We are given the following question. I don't expect a full answer to this question, but I don't have any clue as to what is a Matrix Unit. Do any of you people know what is a matrix unit?
  20. P

    Linear algebra ordered basis problem

    [b]1. The problem statement find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector x = \begin{pmatrix}-1\\-13\\ 9\\ \end{pmatrix} \in\mathbb R if {β= \begin{pmatrix}-1\\4\\ -2\\ \end{pmatrix},\begin{pmatrix}3\\-1\\ -2\\ \end{pmatrix},\begin{pmatrix}2\\-5\\ 1\\ \end{pmatrix}}...
  21. T

    Can you find a basis without deg. 2 polynomials?

    Homework Statement Can you find a basis {p1, p2, p3, p4} for the vector space ℝ[x]<4 s.t. there does NOT exist any polynomials pi of degree 2? Justify fully.Homework Equations The Attempt at a Solution We know a basis must be linearly independant and must span ℝ[x]<4. So intuitively if there...
  22. B

    Some subset of a generating set is a basis

    I'm having some set theoretic qualms about the following argument for the following statement: Let V be a vector space of dimension n and let S be a generating set for V. Prove that some subset of S is a basis for V. The argument is as follows: If ##V = \{ 0 \} ## then it is trivial...
  23. S

    Prove set of sequences is a basis

    Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n. Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i. Show that (e_i), i∈N is a basis for...
  24. Petrus

    MHB Find a Basis for Subspace in P_3(\mathbb{R})

    Hello, Find a basis for subspace in P_3(\mathbb{R}) that containrar polynomial 1+x, -1+x, 2x Also the hole ker T there T: P_3(\mathbb{R})-> P_3(\mathbb{R}) defines of T(a+bx+cx^2+dx^3)=(a+b)x+(c+d)x^2 I am unsure how to handle with that ker.. I am aware that My bas determinant \neq0 well I did...
  25. D

    Proving something to be a basis.

    Homework Statement Letting u=[3, 0, -5], v=[2, 1, 5] and w=[-1, 3, 4], how would I show that a general vector can be written as a linear combination of this 'basis?' Without using an augmented matrix and getting a really messy result by using arbitrary a, b, and c values as the solutions...
  26. M

    Can you switch the basis mid-problem when solving unit mass balance equations?

    Hello, I was wondering if it is allowed when doing problems on multiple unit mass balances to switch the basis made at the beginning of the problem and apply it to a new control volume, while keeping the old basis for previous control volumes? Thank you
  27. S

    Proof: Gromov's short basis, volume comparison

    Hi! I'm having problems understanding the last step of a derivation for a version of a theorem of Gromov's we had in class: In short, the proof takes the Riemannian universal covering (\tilde M, \tilde g, \tilde p) of (M, p) and uses a short basis (\gamma_1, \gamma_2, ...) of the deck...
  28. M

    Vector Analysis - Similarities on Orthonormal Basis

    Homework Statement Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v, |L(v)| = δL * |v| To check that |L(v)| = δL * |v| for all vectors v in principle involves an infinite number of...
  29. C

    What happens to the form basis after making the metric time orthogonal

    Given a basis for spacetime ##\{e_0, \vec{e}_i\}## for which ##\vec{e}_0## is a timelike vector. Of these vectors one can make a new basis for which all vectors are orthogonal to ##\vec{e}_0##. I.e. the vectors $$\hat{\vec{e}}_i = \vec{e}_i - \frac{\vec{e}_i \cdot \vec{e}_0}{\vec{e}_0 \cdot...
  30. D

    Finding a basis for the null space and range of a matrix

    Homework Statement ##S## is a linear transformation and ##\{u_{1},u_{2}\}## is a basis for the vector space. $$ S(u_{1})=u_{1}+u_{2}\\ S(u_{2})=-u_{1}-u_{2} $$ I would like to find a basis of the null space and range of ##S##.Homework Equations In my text, it says that the proper matrix...
  31. B

    Extending the basis of a T-invariant subspace

    Let ##T: V → V ## be a linear map on a finite-dimensional vector space ##V##. Let ##W## be a T-invariant subspace of ##V##. Let ##γ## be a basis for ##W##. Then we can extend ##γ## to ##γ \cup S##, a basis for ##V##, where ##γ \cap S = ∅ ##, so that ## W \bigoplus span(S) = V ##. My question...
  32. evinda

    MHB Construct Orthogonal Basis in R^3: Solve Exercise

    Hello! I am stuck at the following exercise: "Construct an orthogonal basis of R^{3} (in terms of Euclidean inner product) that contains the vector \begin{pmatrix}2\\1 \\-1 \end{pmatrix} " What I've done so far is: Let {(a,b,c), (k,l,m), (2,1,-1)} be the basis. Then since the basis has to...
  33. U

    Matrix Transformation of operator from basis B' to B

    Homework Statement Hi guys, actually this isn't a homework question, but rather part of the working in a textbook on Linear Algebra. Homework Equations The Attempt at a Solution I'm not sure why it's U*li instead of U*il. Shouldn't you flip the order when you do a matrix...
  34. C

    Are vectors assumed to be with respect to a standard basis?

    For example, if were given only a vector <5, 3, 1>, is this assumed to be respect with the standard basis of R^3? And would this mean that any nonstandard basis is with respect to a standard basis?
  35. S

    Basis vectors definition

    Homework Statement A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by \vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}, where it has been emphasized that the basis ε is not necessarily orthonormal...
  36. A

    Finding the basis for a vector space

    Homework Statement Find a basis for the following vector space: The set of 2x2 matrices A such that CA=0 where C is the matrix : 1 2 3 6The Attempt at a Solution I multiplied C by a general 2x2 matrix ...
  37. G

    Topology-bornology as a basis of turbulence

    According to Wikipedia bornology is the minimum amount of structure needed to address boundedness. Topology is the minimum amount of structure needed to address continuity. Topology-bornology (cf. Bornologies and Fuctional Analysis, by Hogbe-Nlend) can be applied to Distributions as bounded...
  38. V

    Polar unit vectors form a basis?

    I keep reading about polar unit vectors, and I am a bit confused by what they mean. In the way I like to think about it, the n-tuple representation of a vector space is just a "list" of elements from the field that I have to combine (a.k.a. perform multiplication) with the n vectors in some...
  39. G

    Is Schwartz Space a Viable Basis for Understanding PDEs?

    Is there a hole in knowledge as to the origins of PDEs? If there is a void, is Schwartz space a suitable basis? Schwartz spaces are intermediate between general spaces and nuclear spaces. Infra-Schwartz spaces are intermediate between Schwartz spaces and reflexive spaces.
  40. T

    Linear Algebra: Basis vs basis of row space vs basis of column space

    In my linear algebra class we previously studied how to find a basis and I had no issues with that. Now we are studying the basis of a row space and basis of a column space and I'm struggling to understand the methods being used in the textbook. The textbook uses different methods to find these...
  41. S

    Preferred basis in Relational Quantum Mechanics

    In RQM all systems are observers. Select the viewpoint with a system S and an observer O. The systema has 2 eigenfunctions |0> and |1> in a basis. Then the evolution from |init>_{O}(|0>+|1>)_{S}. Then the system evolutions to |O0>|0>_{S} or to |O1>|1>_{S} . But how does the measurement select...
  42. G

    Optical rotation and linear basis set

    If I have a 45 degree linear polarized light which I then circularly polarize using a 1/4 wave plate and put this through an optical rotary crystal and then using the equivalent 1/4 wave plate but in the reverse oriention, will I get back a 45 degree linear polarized light? Put another way...
  43. C

    # elements in base does not depend on the basis

    Essentially, I have to show that where {e_1,...,e_n} forms the basis of L, no family of vectors {e'_1 ,..., e'_m} with m>n can serve as the basis of L. The book shows this by saying there exists a 0 vector such that 0 = \sum_{i=1}^{m}x_ie'_i, where not all x_i vanish. I wanted to show it by...
  44. P

    MHB Can the dimension of a basis be less than the space that it spans?

    Let S be a subspace of $\mathbb{R^2}$, such that $S=\{(x,y):2x+3y=0 \}$. Find a basis,$B$, for $S$ and write $u=(-9,6)$ in the $B$ basis. So, I started to solve $2x+3y=0$ for $x$ and I got $x=-\frac{3}{2}y$. Then I could write, $\left[ \begin{matrix} x \\ y \end{matrix}\right] = \left[...
  45. S

    Finding a Basis for a set of vectors

    Homework Statement Let H be the set of all vectors of the form (a-3b, b-a, a, b) where a and b are arbitrary real scalars. Show that H is a subspace of ℝ^4 and find a basis for it. Right, I've shown it's a proper subspace, just need help with finding a basis. Is {a-3b, b} a suitable...
  46. Math Amateur

    MHB Hilbert's Basis Theorem - Polynomial of Minimal Degree

    I am reading the Proof of Hilbert's Basis Theorem in Rotman's Advanced Modern Algebra ( See attachment for details of the proof in Rotman). Hilbert's Basis Theorem is stated as follows: (see attachment) Theorem 6.42 (Hilbert's Basis Theorem) If R is a commutative noetherian ring, the R[x] is...
  47. Math Amateur

    MHB Hilbert's Basis Theorem - Basic Question about proof

    I am reading Dummit and Foote Section 9.6 Polynomials In Several Variables Over a Field and Grobner Bases I have a very basic question regarding the beginning of the proof of Hilbert's Basis Theorem (see attachment for a statement of the Theorem and details of the proof) Theorem 21...
  48. M

    Physical basis for high-bypass turbofans

    Hi, Can someone double check I understand this correctly? The turbofan has lower specific fuel consumption because a gas's momentum is proportional to its velocity, whereas a gas's kinetic energy is proportional to its squared velocity. Therefore a turbojet can be made more efficient by adding...
  49. F

    LINEAR ALGEBRA: image of vectors through other basis

    Homework Statement In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)## and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where (1 0 0) (0 2 0) = A (0 0 0)...
  50. O

    Finding Basis & Spanning Set for Matrix: a,b,c,d

    I'm having trouble finding the spanning set and basis for the matrix; | a b | | c d | with condition that b=d I'm thinking thinking the spanning set would be A= x B = y C = z Such that x,y,z are all reals, but I can't think of how to find a basis for this, I'm thinking of doing...
Back
Top