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. P

    Writing Non-Coordinate Basis Continuity Eqs: $\nabla_a T^{ab}$

    The continuity equations are ##\nabla_a T^{ab} = 0##. In a coordinate basis, we can write out the resulting differential equations as: ##\nabla_a T^{ab} = \partial_a T^{ab} + \Gamma^a{}_{ac}T^{cb} + \Gamma^b{}_{ac}T^{ac}## (modulo possible typos, though I tried to be careful-ish). What do we...
  2. raay

    Linear Algebra - Change of Basis

    Hi please i need help in number 3 of the tutorial questions. It is not an assignment its just a tutorial (read title in the image). I am currently studying for my final and i need help in (3b). the only way I am thinking of solving this questions is to use the equation given in part (d). But...
  3. RyanH42

    Basis and Components (Vectors)

    Homework Statement Let ##u_1,u_2,u_3## be a basis and let ##v_1=-u_1+u_2-u_3## , ## v_2=u_1+2u_2-u_3## , ##v_3=2u_1+u_3## show that ##v_1,v_2,v_3## is a basis and find the components of ##a=2u_1-u_3## in terms of ##v_1,v_2,v_3## Homework Equations For basis vecor...
  4. B

    Vector Space Basis: Clarifying Linear Independence

    Hi Folks, I find this link http://mathworld.wolfram.com/VectorSpaceBasis.html confusing regarding linear independence. One of the requirement for a basis of a vector space is that the vectors in a set S are linearly independent and so this implies that the vector cannot be written in terms of...
  5. D

    Coordinate charts and change of basis

    So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...
  6. S

    Figuring out Bravais lattice from primitive basis vectors

    Homework Statement Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are...
  7. ELB27

    Completing to a basis

    Homework Statement Consider in the space ##\mathbb{R}^5## vectors ##\vec{v}_1 = (2,1, 1, 5, 3)^T## , ##\vec{v}_2 = (3, 2, 0, 0, 0)^T## , ##\vec{v}_3 = (1, 1, 50, 921, 0)^T##. a) Prove that these vectors are linearly independent. b) Complete this system of vectors to a basis. If you do part b)...
  8. K

    Lattice points and lattice basis

    Hi! I'm struggling in identifying the lattice points and atom basis. As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct? So if the points on the corners are the lattice...
  9. 1

    Rewrite state in new basis - Quantum Mechanics

    Homework Statement Rewrite the state |ψ⟩ = √(1/2)(|0> + |1>) in the new basis. |3⟩ = √(1/3)|0⟩ + √(2/3)|1⟩ |4⟩ = √(2/3)|0⟩ − √(1/3)|1⟩ You may assume that |0⟩ and |1⟩ are orthonormal. Homework Equations The Attempt at a Solution [/B] I have a similar example in my notes however there...
  10. R

    What the terms orthogonal & basis function denote in case of signals

    I am a beginer. I have read that any given signal whether it simple or complex one,can be represented as summation of orthogonal basis functions.Here, what the terms orthogonal and basis functions denote in case of signals? Can anyone explain concept with an example?Also,what are the physical...
  11. R

    Algorithm to compute Basis images of an image

    I know from the Fourier Analysis that any signal can be represented as summation of elementary signals i.e. basis functions .Likewise,any image can be represented as summation of Basis images. Is there any available code, or even an algorithm, that would allow me to compute Basis images of an...
  12. K

    MHB Basis Theorem for Finite Abelian Groups

    I am attempting to answer the attached question. I have completed parts 1-4 and am struggling with part 5. 5. Prove that if a^{l_0}b_1^{l_1}...b_n^{l_n}=e then a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e If |a|>|b1|>|b2|>...>|bn| then I could raise both sides of a^{l_0}b_1^{l_1}...b_n^{l_n}=e to the...
  13. Adoniram

    How to write this state vector in coordinate basis?

    Homework Statement I am given this state, which is the result of a lamba particle decaying into a proton and neutral pion. Initial j = 3/2. The final state can theoretically be written as: I have already determined that: alpha_p = Sqrt[2/3] beta_p = Sqrt[1/3] alpha_d = -/+ Sqrt[2/5]...
  14. B

    How do you get a matrix from this basis?

    Homework Statement Here's my problem. I only need help with the bottom part, but if you could explain the problem more vividly that would help too. Homework Equations A = S-1BS (?) There aren't really any relevant equations. This part of linear algebra is getting really abstract, at least I...
  15. K

    Show that the Slater Determinant states are a complete basis

    Homework Statement "Show that the Slater Determinant states are a complete basis" is the entire statement. Homework EquationsThe Attempt at a Solution I guess I'm trying to prove that the rank of the states is equal to the basis? I'm not sure where to start on this one.
  16. P

    Given the basis of find the matrix

    Homework Statement Not a homework problem. Typically, we are given a matrix, then asked to find the basis for the kernel or image space of the matrix. I've never seen a problem that did the converse (i.e., given the matrix for the kernel/image space of some matrix, find some matrix). I was...
  17. Dethrone

    MHB Finding a basis for a Vector Space

    4b). How can I find a basis? I was thinking of the standard basis $\{1,x,x^2\}$, but that doesn't work under the scalar multiplication definition in the vector space. EDIT: I think it is $\{0,x,x^2\}$ and we take $1$ to be the $0$ vector! $a(0)+b(x)+c(x^2)=1$ implies $a=b=c=0$. It is strange...
  18. L

    Effects of Hamiltonian Preferred Basis on Decoherence

    What would be the effects on the system for different values of the Hamiltonian preferred basis in Decoherence? Would it for example make the electrons higher in orbital or bands? Or what would be the exact effects?
  19. D

    Transforming Vectors from Basis B to C: A Confusing Matter

    Suppose a change of basis from basis ##B## to basis ##C## is represented by the matrix ##S##. That is, ##S## is the transformation matrix from ##B## to ##C##. Now if ##t## is a given linear transformation, ##t:~V\rightarrow V##, with eigenvectors ##\epsilon_i##, say, and ##T## is the...
  20. K

    Basis of the solution space of a differential equation

    Homework Statement Verify that the functions y1(x) = x and y2(x) = 1/x are solutions of the differential equation y'' + (1/x)y' - (1/x2)y = 0 on I = (0,∞). Show that y1(x), y2(x) is a basis of the solution space of the differential equation. The Attempt at a Solution For the first part I'll...
  21. F

    Find basis B given the transition matrix and B'

    Homework Statement The Matrix P = 1 0 3 1 1 0 0 3 1 is the transition matrix from what basis B to the basis B' = {(1,0,0),(1,1,0),(1,1,1) for R3? Homework Equations [v]B=P[v]B' The Attempt at a Solution I'm looking...
  22. amjad-sh

    Understanding Inner Product in Infinite Dimensional Bases

    While I'm reading a book in quantum mechanics, I reached the part "Generalization to infinite dimension". We know that at infinite dimension many definitions changes.And that what is confusing me! Take for example the inner product.when we are dealing in finite dimension the definition of inner...
  23. K

    Find a basis of a linear system

    Find a basis for the solution space of the linear system x1-x2-2x3+x4 = 0 -3x1+3x2+x3-x4 = 0 2x1-2x2+x3 = 0 I created a matrix (not augmented, will be 0 on right side no matter what row operations) and brought it to reduced echelon form. x2 and x4 were free variables and I set them to the...
  24. PcumP_Ravenclaw

    Abstract Vector Basis: Necessary & Sufficient Condition for Plane Representation

    Homework Statement Suppose that ## u = s_1i + s_2j ## and ## v = t_1i + t_2j ##, where s1, s2, t1 and t2 are real numbers. Find a necessary and sufficient condition on these real numbers such that every vector in the plane of i and j can be expressed as a linear combination of the vectors u and...
  25. curiousman

    Medical Alternative Medicine - Any scientific basis?

    Dear all, I have been searching some scientific basis about the most known -there are others less popular- complementary & alternative medicines listed below: - Aromatherapy - Ayurvedic Medicine - Bach Flowers - Chiropractic - Chromotherapy - Iridiology - Kinesiology - Oligotherapy -...
  26. A

    MHB Basis for column and null space

    Please help me with these three questions. I'm really struggling to understand these concepts and I think that with an understanding of these three, I will be able to tackle the rest before my test on Wednesday. Thank you. http://www.texpaste.com/n/g4rwmzzw 1) $$ A = \left[\begin{matrix} -6 &...
  27. I

    Basis of a Vector Space

    Homework Statement Let V be a vector space, and suppose that \vec{v_1}, \vec{v_2}, ... \vec{v_n} is a basis of V. Let c\in\mathbb R be a scalar, and define \vec{w} = \vec{v_1} + c\vec{v_2}. Prove that \vec{w}, \vec{v_2}, ... , \vec{v_n} is also a basis of V. Homework Equations If two of the...
  28. K

    Prime Numbers as Ortho-normal basis for all numbers

    Hi, Can we treat prime numbers as an Ortho-normal basis of "Infinite" dimensions to represent every possible number. Treating numbers as vectors. Thanks.
  29. B

    Basis and Dimension of Solution Space

    Homework Statement Find a basis for and the dimension of the solution space of the homogenous system of equations. 2x1+2x2-x3+x5=0 -x1-x2+2x3-3x4+x5=0 x1+x2-2x3-x5=0 x3+x4+x5=0 Homework EquationsThe Attempt at a Solution I reduced the vector reduced row echelon form. However the second row...
  30. A

    Finding the orthonormal basis for cosine function

    Homework Statement si(t) = √(((2*E)/T)*cos(2*π*fc*t + i*(π/4))) for 0≤t≤T and 0 otherwise. Where i = 1, 2, 3, 4 and fc = nc/T, for some fixed integer nc. What is the dimensionality, N, of the space spanned by this set of signal? Find a set of orthonormal basis functions to represent this set of...
  31. B

    Test Tomorrow Linear independence, spanning, basis

    Hello I'm taking linear algebra and have a couple of questions about linear independence, spanning, and basis Let me start of by sharing what I think I understand. -If I have a matrix with several vectors and I reduce it to row echelon form and I get a pivot in every column then I can assume...
  32. ichabodgrant

    Finding a basis of a space

    Let u = [1, 2, 3, -1, 2]T, v = [2, 4, 7, 2, -1]T in ℝ5. Find a basis of a space W such that w ⊥ u and w ⊥ v for all w ∈ W. I think the question is quite easy. Given this vector w in the space W is orthogonal to both u and v. I can only think of w being a zero vector. But would this be too...
  33. R

    Basis Atom of Superconductor

    Hi everyone, I'm trying to simulated the XRD pattern of Bi2Sr2Ca2Cu3O10, but I'm having a problem of finding the basis of atom(and their respective position). Also its JCPDS is quite hard to find, so if anyone working with this, may you provide a link or articles about my problem. Thanks...
  34. R

    Matrix representation in x and y basis for spin operators

    Homework Statement How can I find the matrix representation of ##\mathbb{S}_+## and ##\mathbb{S}_-## in the ##|\pm y\rangle## or ##|\pm x\rangle## basis?Homework Equations ## \mathbb{\hat{S}}_+|s,m\rangle = \sqrt{s(s+1)-m(m+1)}\hbar|s,m+1\rangle ## The Attempt at a Solution The book almost...
  35. S

    MHB Faces of a simplex, convex hull of the canonical basis of \$\mathbb{R}^n\$

    Let $I_n := \{1,2,...,n \}, \ p \in \Delta_n = \{(p_1, ..., p_n) \ | \ p_i \ge 0, \sum_{i=1}^n =1\}$ $ \text{supp (p)}= \{ i \in I_n \ | \ p_i \neq 0\}$ For a convex set $C$ we define $F$ to be its face if $F$ is convex and $\forall x,y \in C, \lambda \in (0,1) : \lambda x + (1- \lambda ) y...
  36. Dethrone

    MHB Find a Basis for U with $\text{rref}(A)$

    Find a basis for $U=\text{span}{}\left\{\begin{bmatrix}1\\1\\0\\0\end{bmatrix}\begin{bmatrix}0\\0\\1\\1\end{bmatrix}\begin{bmatrix}1\\0\\1\\0\end{bmatrix}\begin{bmatrix}0\\1\\0\\1\end{bmatrix}\right\}$ Let $U=\text{col}(A)$, and applying row reduction on A, we obtain...
  37. jk22

    Non-separability : basis dependent

    If we consider the singlet state (0,1,-1,0)/sqrt2 then it is easy to see that the unitary block transformation : A=RoR^-1with R a rotation of 45 degrees gives the vector 1/2(-1,1,-1,1) which is separable. Thus entanglement disappears in that basis.
  38. kini.Amith

    Separation of variables to solve Schrodinger equation

    How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...
  39. PsychonautQQ

    Splitting field, prime basis

    1. The problem statement, all variafbles and given/known data Problem: Let E be a splitting field of f over F. If [E:F] is prime, show that E=F(u) for some u in E (show that E is a simple extension of F) Homework Equations Things that might be useful: If E>K>F are fields, where K and F are...
  40. R

    Euclidean space: dot product and orthonormal basis

    Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...
  41. Ibix

    Coordinate basis for cotangent space

    Warning: this may be totally trivial, or totally wrong. I've been working through Sean Carroll's lecture notes, and I've got to http://preposterousuniverse.com/grnotes/grnotes-two.pdf . I follow the derivation for showing that the tangent space bases are the partial derivatives (Carroll's...
  42. A

    Understanding Basis Change with Hamiltonian Matrices

    We are given the vectors la> = (1,0) and lb> = (0,1) and then a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entires and zero elsewhere. I am asked to now represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H...
  43. Y

    MHB Extending Vectors to a Basis of R^4: Why Notation Matters

    Hello all I am trying to solve this problem: Extend the following vectors to a basis of R^4. \[u_{1}=\left ( \begin{matrix} 1\\1 \\1 \\1 \end{matrix} \right )\] and \[u_{2}=\left ( \begin{matrix} 2\\2 \\3 \\4 \end{matrix} \right )\] What I did, I put these vectors as columns of a matrix...
  44. Yangyin

    The basis for all things that exist

    what is the basis for all things that exist ? does it start with a single atom? are there atoms that exist in every thing in existence ? how are they different and how are they the same ?
  45. K

    Deducing Basis of Set T from Coordinates in Matrix A with Respect to Basis S

    Hello, I am just doing my homework and I believe that there is a fault in the problem set. Consider the set of functions defined by V= f : R → R such that f(x) = a + bx for some a, b ∈ R It is given that V is a vector space under the standard operations of pointwise addition and scalar...
  46. C

    Easy question regarding the basis for a topology

    Hello, I know that given a set $X$ and a topology $T$ on $X$ that a basis $B$ for $T$ is a collection of open sets of $T$ such that every open set of $T$ is the Union of sets in $B$. My question is: does taking the set of all Unions of sets in $B$ give exactly the topology $T$ ?
  47. P

    Matrix representation of an operator in a new basis

    Homework Statement Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis. Homework EquationsThe Attempt at a Solution...
  48. M

    Subspace / basis problem

    Homework Statement Let S, a subspace of ℝ3 be the set of vectors orthogonal to vector (1,2,3) a)describe Set S b) find a basis for Set S 2. Relevant Equations That a basis has to be linearly independent and span R^3The Attempt at a Solution [/B] I would do this: I know that vector (1,2,3) is...
  49. M

    Difference between span and basis

    I'm just having a small trouble understanding the difference ( occurred while I was doing exercise). A basis is defined as 1)linearly independent 2)spans the space it is found in. Here is where I get confused: To determine whether or not a set spans a vector space, I was taught to find its...
  50. K

    Is it possible to determine a basis for non-subspace spaces?

    The title may seem a little confusing and possibly stupid :D What I mean is like a plane doesn't go through the origin? Can we describe a basis for this? If so, how?
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