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. C

    Finding a Basis for the Nullspace of a 2x2 Matrix Transformation

    The problem is attached. I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1Applying the transformation to each of these gives 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 respectively. Now this is where I get stuck. How do I find a...
  2. C

    Basis for Range of Linear transformation

    The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...
  3. C

    Finding basis for nullspace of transformation

    T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...
  4. Mentz114

    Transformation of an acceleration vector under a basis change

    This thread is spawned from an earlier one https://www.physicsforums.com/showthread.php?t=647147&page=7 For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\...
  5. C

    Can Linear Independence Be Determined Without Row Reduction?

    Suppose that S = {v1, v2, v3} is a basis for a vector space V. a. Determine whether the set T = {v1, v1 + v2, v1 + v2 + v3} is a basis for V. b. Determine whether the set W = {−v2 + v3, 3v1 + 2v2 + v3, v1 − v2 + 2v3} is a basis for V. I must check if they're linearly independent...
  6. P

    Subspace & Basis: Proving and Understanding

    The problem is attached. I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right. Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions: 1) 0 is in S 2) if U and...
  7. P

    Efficient Method for Finding Basis and Determinant of 4 Vectors in Matrix Form

    I know how to do the problem, just put the 4 vectors in matrix form and find for what values of k is the detminant =0. the answer is then that k can't equal the value that was found. Is there a easier way to do this? My method involves finding the determinant using the expansion method...
  8. C

    Proving the Basis Property of S' for Rn Using Invertible Matrices

    Show that if S = {v1, v2, . . . , vn} is a basis for Rn and A is an n × n invertible matrix, then S' = {Av1,Av2, . . .,Avn} is also a basis. I need to show that: 1) Av1, Av2,...Avn are linearly independent 2) span(S)=Rn I'm having some problems with this. I see that S'=AS (duh)...
  9. C

    Find Basis for diagonal matrix

    I'm not sure how to start this problem. All i know is a diagonal matrix consists of all 0 elements except along the main diagonal. But how do I even find a basis for this?
  10. Y

    MHB Basis & Dimension of 2x2 Matrix Subspaces: W1 & W2

    Hello I have this problem, I find it difficult, any hints will be appreciated... Two subspaces are given (W1 and W2) from the vector space of matrices from order 2x2. W1 is the subspace of upper triangular matrices W2 is the subspace spanned by...
  11. A

    Infinite dimensional vector spaces without basis?

    According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis...
  12. B

    Completeness of a set of basis vectors in 3D Euclidean space.

    Homework Statement The problem is Exercise 2 in the picture http://postimage.org/image/3ou3x1sh7/ Homework Equations The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors? The Attempt at a Solution I have no idea where...
  13. N

    Find the matrix representations of the Differentiation Map in the Basis

    Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...
  14. N

    What is the meaning of a basis?

    Hi! There is a concept I don't understand and would love to have is cleared... What is the meaning of a lattice with a basis? What do I need it for? Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? ) why does it become...
  15. P

    Basis for Matrix: Is [0 0]^t Always the Answer?

    When you have a matrix like: 3 1 0 1 The RREF is 1 0 0 1 the identity matrix. Is the basis always [0 0]^t?
  16. P

    Do Linear Operators Equate on All Vectors if They Match on a Basis Set?

    Suppose that T1: V → V and T2: V → V are linear operators and {v1, . . . , vn} is a basis for V . If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show that T1(v) = T2(v) for all v in V . I don't understand this question. They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n...
  17. N

    Finding the Dimension and Basis of the Matrix Vector space

    Homework Statement The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations. It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set...
  18. D

    How to Represent |+x> and |-x> Using |+y> and |-y> as Basis?

    Homework Statement Determine the column vectors representing the states |+x> and |-x> using the states |+y> and |-y> as a basis. Homework Equations ? The Attempt at a Solution The hint my prof gave us was that since |+x> = 1/√2|+z> + 1/√2|-z> we can eliminate the states |+z> and...
  19. A

    How to find a basis for the vector space of real numbers over the field Q?

    So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers...
  20. F

    Dot product between arrays: basis representation of an image

    Hello Forum, When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors: x= a1 v1 + a2 v2 + a3 v3+ ... Each coefficient a_i is the dot product between x and each basis vector v_i. If the vector x is not a row (or column vector)...
  21. D

    Proving Existence of Basis in Vector Spaces

    Hi, I've been trying to prove that every vector space has a basis. So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of...
  22. B

    How Do You Write the Wavefunction for Two Electrons in Quantum States?

    Homework Statement This is something I should know, but I keep getting mixed up when I try to think about it. A quantum state can be written as a superposition of basis states such as \left | n \right \rangle So let's say I have a particle in a potential with discrete energy levels...
  23. B

    Is it possible to define a basis for the space of continuous functions?

    In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions? I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...
  24. P

    Is matrix A in the form of RREF?Is Matrix A in RREF form?

    I attached 2 problems. For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get x1=-2t x2=-t x3=s x4=u (using u because I'm using t to denote transpose) where x1 to x4 is for each respective column...
  25. A

    A question about linear algebra (change of basis of a linear transformation)

    Homework Statement Let A \in M_n(F) and v \in F^n. Let v, Av, A^2v, ... , A^{k-1}v be a basis, B, of V. Let T:V \rightarrow V be induced by multiplication by A:T(w) = Aw for w in V. Find [T]_B, the matrix of T with respect to B. Thanks in advance Homework Equations...
  26. D

    Understanding basis and dimension

    I am really confused about something. I know that if I have a vector space, then the dimension of that vector space is the number of elements in a basis for it. But this brings up some confusing issues for me. For example, if we are looking at the null space of a non-singular, square matrix...
  27. ElijahRockers

    Find a basis for a set S of R4

    Homework Statement Find a basis for the subspace S of vectors (A+B, A-B+2C, B, C) in R4 What is the dimension of S? The Attempt at a Solution Do I just plug in varying values for A B and C to create four vectors, and see if they are linearly independent? If they are then I've found...
  28. N

    The basis of the Real numbers over the Irrationals

    1. What can be said of the dimension of the basis of the Reals over the Irrationals 2. Homework Equations 3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number...
  29. S

    Lineal Transformation basis change

    Find http://imageshack.us/a/img35/1637/lineal2.gif http://imageshack.us/a/img210/1370/lineal1.gif C^3 is the canonical base of ℝ^3, C^2 is the canonical base of ℝ^2 I tried: http://imageshack.us/a/img822/6274/lineal3.gif But I'm not sure if this is right, I made a...
  30. T

    Finding a basis for a set of polynomials and functions

    Find a basis for and the dimension of the subspaces defined for each of the following sets of conditions: {p \in P3(R) | p(2) = p(-1) = 0 } { f\inSpan{ex, e2x, e3x} | f(0) = f'(0) = 0} Attempt: Having trouble getting started... So I think my issue is interpreting what those sets...
  31. K

    Non-coordinate basis for vector fields

    Hello! Im trying to read some mathematical physics and have problems with the understanding of vector fields. Th questions are regarding the explanations in the book "Geometrical methods of mathematical physics".. The author, Bernard Schutz, writes: "Given a coordinate system x^i, it is often...
  32. A

    How Do You Transform Two-Electron Integrals from Atomic to Molecular Basis?

    Hello Everyone, Can anyone tell me how to do a transformation from atomic basis to molecular basis incase of 2-electron integrals for the case of Gaussian type Orbitals? Like in case of One electron integrals it is known that a transformation from the Gaussian type orbitals to their...
  33. fluidistic

    Completing the basis of a matrix, Jordan form related

    Hi guys, Let's say I have a 6x6 matrix A whose Jordan form J has 3 Jordan blocks. It means that this matrix (matrix A, but I think that also the matrix J) has 3 linearly independent eigenvectors, I have no problem in finding them. I simply do (A-\lambda _i I)v_i=0 to get the eigenvectors v_i...
  34. C

    Basis of a subspace and dimension question

    How do i go about this? Find a basis for the subspace W of R^5 given by... W = {x E R^5 : x . a = x . b = x . c = 0}, where a = (1, 0, 2, -1, -1), b = (2, 1, 1, 1, 0) and c = (4, 3, -1, 5, 2). Determine the dimension of W. (as usual, "x . a" denotes the dot (inner) product of the...
  35. H

    Understanding Basis Vectors and One-Forms: A Simplified Explanation

    Greetings, I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as: e\mu = \partial/\partialx\mu. I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a...
  36. Alesak

    Don't understand continuous basis

    Hi, I'm beginning to learn QM, and I've never seen any treatment of vector spaces with infinite bases. Countable case is quite digestible, but uncountable just flies over my head. Can anyone recommend me place where to learn this more advanced part of linear algebra, with focus on stuff...
  37. O

    What do engineers do on a daily basis?

    I am pretty sure I want to go into engineering, but I am really curious as to what engineers do on a daily basis. I have this vague and somewhat childish idea that it is just a bunch of people in overalls tinkering with machine parts. That just shows how little I really know. I'm looking for...
  38. C

    Finding a basis of eigenvectors

    Homework Statement A = \left( \begin{array}{ccc} 2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & -1 \end{array} \right) Find the eigenvalues and corresponding eigenvectors that form a basis over R3 Homework Equations The Attempt at a Solution OK so I've found the characteristic...
  39. J

    Linear transformation and Change of Basis

    Homework Statement Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance. If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and T(v2)=3v2. and B =...
  40. V

    Basis for an irreducible representation

    Homework Statement I'm having trouble understanding a concept in representation theory. I've been reading several texts on the application of rep. theory to quantum mechanics ("Group Theory and Quantum Mechanics" by Tinkham and "Group Theory and Its Application to Physical Problems" by...
  41. C

    What Basis Spans Square Integrable Functions with Exponential Tails on [0,∞)?

    Hi, I have a function on [0,\infty) which is represented as: \sum_{ \stackrel{ \Re( \alpha )\in\mathbb{Q}^+ }{ \Im( \alpha )\in\mathbb{Q} } }{\beta_\alpha e^{-\alpha t}} It seems like this must be a basis for the square integrable functions on [0,\infty) with exponential tails. Am I right...
  42. G

    Find a basis of a subspace of R^4

    Find a basis of the given span{[2,1,0,-1],[-1,1,1,1],[2,7,4,5]} So I got the RREF, and found the basis to be two rows of the RREF, which are [1,0,-1/3,0] and [0,1,2/3,1], but the answer is [2,1,0,-1],[-1,1,1,1] Where did I do wrong?
  43. M

    I am not sure - a manifold is locally connected and has countable basis?

    I am not sure -- a manifold is locally connected and has countable basis? There is an Exercise in a book as following : Given a Manifold M , if N is a sub-manifold , an V is open set then V \cap N is a countable collection of connected open sets . I am asking why he put this exercise...
  44. matqkks

    Unlock the Power of Basis Vectors: Impactful Examples

    I have normally introduced basis vectors by just stating independent vectors that span the space. This is perhaps not very inspirational. What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. Maybe a good...
  45. matqkks

    MHB How Can Basis Vectors Simplify Real-World Problems?

    What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. I have normally introduced it by just stating independent vectors that span the space.
  46. M

    Riemann tensor cyclic identity (first Bianchi) and noncoordinate basis

    I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...
  47. A

    What Am I Missing in Change of Basis Matrices?

    Homework Statement https://dl.dropbox.com/u/4788304/Screen%20shot%202012-07-08%20at%2002.53.44.JPG This is the solution of Problem A.15 in Griffiths' Quantum Mechanics. Tx is the rotation matrix about x-axis for theta degrees; while Ty is the rotation matrix about y-axis for theta degrees...
  48. M

    Modern physics basis for magnetic properties of matter

    Do you people know any decent and simple (no fancy math) material for the atomic physics around diamagnetism, paramagnetism and ferromagnetism? Topics like wave-particle duality, energy levels, solids, why metals are conductors, relation between magnetic moment and effective mass of...
  49. PerpStudent

    Is there an intuitive basis for the Lagrangian?

    Since it is based on the kinetic energy less the potential energy, what does the Lagrangian actually represent? Is there some intuitive way to understand why it is defined so and why it is such a fruitful concept using the principle of least action?
  50. O

    Linear Algebra - Jordan form basis

    Hi all, I'm having trouble finding jordan basis for matrix A, e.g. the P matrix of: J=P^{-1}AP Given A = \begin{pmatrix} 4 & 1 & 1 & 1 \\ -1 & 2 & -1 & -1 \\ 6 & 1 & -1 & 1 \\ -6 & -1 & 4 & 2 \end{pmatrix} I found Jordan form to be: J = \begin{pmatrix} -2 & & & \\ & 3 & 1 & \\ & & 3 &...
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