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. P

    Calculating G' for an Orthogonal Coordinate System

    Homework Statement For the orthonormal coordinate system (X,Y) the metric is \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} Calculate G' in 2 ways. 1) G'= M^{T}*G*M 2) g\acute{}_{ij} = \overline{a}\acute{}_{i} . \overline{a}\acute{}_{j} Homework Equations \begin{pmatrix}...
  2. D

    Basis for Null Space: Finding Basis Vectors | Explained and Solved

    Hey guys so we need to find the basis for 0 1 \sqrt{2} 0 0 0 0 0 0 I know how you do it. But my prof says that one of the basis vectors is (1 0 0) but I don't know how he arrives at this?
  3. 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...
  4. 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...
  5. 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 \(u_1=b_1+b_2+b_3+b_4+b_5\)...
  6. 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 -...
  7. N

    How Do You Determine a Basis for a Set of Vectors?

    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...
  8. Sudharaka

    MHB Can Any Bivector Be Decomposed Using a Specific 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...
  9. 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 $e_i\Lambda e_j$...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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}...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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)}...
  21. Sudharaka

    MHB Understanding Matrix Units 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?
  22. 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}}...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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?
  37. S

    How Do You Normalize Basis Vectors in Non-Orthogonal Systems?

    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...
  38. 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 ...
  39. 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...
  40. 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...
  41. 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.
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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[...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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