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).
Hi all,
This is both linear algebra and physics problem, and I decided to post in physics because I want a "physics-framed" answer.
Suppose you have a system with two objects (subsystems) in it described by the state:
|ψ> = ƩiƩjcij|i>|j>
where |i> and |j> are orthonormal bases for the two...
If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...
Hi there!
A Hilbert space E is spanned by a set S if E is generated by the element of S.
It is well known that in the finite dimensional case that
S spans E and S is linearly independent set iff the set S form a basis for E.
The question is that true for the infinite dimensional...
Homework Statement
Let W = \begin{cases} \begin{pmatrix}x\\y\\z\\w\end{pmatrix} \in R^4 | w + 2x + 2y + 4z = 0 \end{cases}
A)Find basis for W.
B)Find basis for W^{\perp}
C)Use parts (A) and (B) to find an orthogonal basis for R^4 with
respect to the Euclidean inner product.
Homework...
If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was...
Homework Statement
Homework Equations
...
The Attempt at a Solution
Can someone just point me how to approach this? Do we take a random second degree polynomial and input 2x + 3 instead of x, then find the constants (eg. denoted by a , b , c) by putting the new equation equal to...
I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...
Homework Statement
V = {p(x) belongs to P3 such that p'(1) + p'(-1) = 0}
Homework Equations
...
The Attempt at a Solution
Okay, so finding the first derivative of p(x) = ax^3 + bx^2 + cx + d and plugging in the values 1 and -1 (to find p'(1) and p'(-1)), we get c = -3a. Does this make the...
Homework Statement
$$
\begin{pmatrix}
-1&3&0\\
2&0&-1\\
0&-6&1
\end{pmatrix}
$$
Finding the ImT basis of this
The Attempt at a Solution
I got it down to
$$
\begin{pmatrix}
1&0&-1/2\\
0&1&1/6\\
0&0&1
\end{pmatrix}
$$
I know that by the principle of having pivots as the only non-zero...
Homework Statement
The problem along with its solution is attached as Problem 1-2.jpg.
Homework Equations
Norm of a vector.
The Attempt at a Solution
Starting from the final answer of the solution, sqrt((-0.625)^2 + (0.333)^2) == 0.708176532 != 1. Did the book do something wrong? I ask...
I was wondering whether we can use row as well as column operations to reduce a matrix to find column space? Or do we only have to perform row operations to reduce matrix in case of row space and column operations to find column space?
Homework Statement
w1 = 2 1 2
w2 = 1 -2 -3
w3 = 5 0 1
for
R3
Homework Equations
The Attempt at a Solution
The books says the above is not a basis, why not? There are no free variables, none of the vectors are multiples of the other, they are linearly independent and the...
Im having trouble under stand the relationships between determinats, span, basis.
Given a 3x3 matrix on R3 vector space.
* If determinat is 0, it is linearly dependent, will NOT span R3, is NOT a basis of R3.
, If determinant is non-zero, its linearly independent, will span R3, is a basis of...
From what I understand, a basis is essentially a subset of a vector space over a given field.
Now what I'm not so sure of is the linearly independence part. If the basis has two linearly independent vectors, then than means they aren't collinear: rather, they wouldn't have the same slope...
I've just studied the implicit function theorem and if we assume the theorem is true then we can easily compute the following:
id_n = D(id_n)_x = D({f^-1}°{f})_x = {D(f^-1)_f(x)} ° {Df_x}
where D(*)_a means the derivative of * at x.
OKay... so this was very straightforward until I began...
Hi There,
I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a...
Homework Statement
Is the system of vectors a1=(1,-1,0,1), a2=(2,3,-1,0), a3=(4,1,-1,4) linearly independent? Do these vectors form a basis in the vector space R^4? State why.
Homework Equations
The Attempt at a Solution
I have done the first part of the exercise. I have found...
Homework Statement
Let A be an 3x3 matrix so that A^3 = {3x3 zero matrix}. Assume there is a vector
v with [A^2][v] ≠ {zero vector}.
(a) Prove that B = {v; Av; [A^2]v} is a basis.
(b) Let T be the linear transformation represented by A in the stan-
dard basis. What is [T]B?
Homework...
Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in Fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just...
First of all, I'd like to say hi to all the peole here on the forum!
Now to my question:
When reading some general relativity articles, I came upon this strange notation:
T^{a}_{b} = C(dt)^{a}(∂_{t})_{b} + D(∂_{t})^{a}(dt)_{b}. Can someone please explain to me what this means? Clearly...
Hi there, I am reading the book "Condensed Matter Physics" second edition by Michael P. Marder. It stated in page 9 that one basis of the the honeycomb lattice is
\vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad
\vec{v}_2 = a [0 \ -1/(2\sqrt{3})]
which is based on figure 1.5(B) in page...
Find a basis and the dimension for the subspace of R^3(3D) spanned by the vectors {(0,1,-2),(3,0,1),(3,2,-3)}
The dimension is 2 regardless if i put the vectors in row space or column space form.
But to find the basis I need to put it in row space form.
Can anyone please explain when I...
The set is (1,3),(-1,2),(7,6)
it is in R2 so I don't get why there are 3 elements.
I assumed they are not vectors but points instead.
but if they are points then it becomes a line, and the answer is that its dimension is 2, and a basis is (1,0) and (0,1)
Could someone explain this?
Thanks
http://dl.dropbox.com/u/33103477/linear%20transformations.png
My solution(Ignore part (a), this part (b) only)
http://dl.dropbox.com/u/33103477/1.jpg
http://dl.dropbox.com/u/33103477/2.jpg
So I have worked out the basis and for the kernel of L1 and image of L2, so I have U1 and U2...
I am working on a problem dealing with transformations of a vector and finding the basis of its kernel. Now I have worked out everything below but after reading the definitions I am a bit confused, hence just want verification if the procedure I am following is correct.
My transformed matrix is...
http://dl.dropbox.com/u/33103477/linear%20transformations.png
My attempt was to first find the transformed matrices L1 and L2.
L1= ---[3 1 2 -1]
-------[2 4 1 -1]
L2= ---[1 -1]
-------[1 -3]
-------[2 -8]
-------[3 -27]
Now reducing L1, I have
-------[1 0 7/10 -3/10]...
Hi everyone,
Pardon the neophyte question, but is a one-form the same thing as a dual basis vector? If not, are they related in some way, or completely different concepts/entities?
Thank you!
B&F have the following:
\delta_{i j} = e'_i \cdot e'_j = a_{i k} \left( e_k \cdot e'_j \right) = a_{i k} a_{j k}
and they ask the reader to show that
a_{k i} a_{k j} = \delta_{i j}
Does it suffice to show the following? :
\delta_{i j} = a_{i k} a_{j k} \to \delta_{j i} = \left(...
How the hell do you prove that the components of a vector w.r.t. a given basis are unique?
I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!
hello :)
I was trying to prove the following result :
for a linear mapping L: V --> W
dimension of a domain V = dimension of I am (L) + dimension of kernel (L)
So, my doubt actually is that do we really need a separate basis for the kernel ?
Theoretically, the kernel is a subspace of the...
Homework Statement
Let W be a subspace of ℝ4 spanned by the vectors:
u1 = [1; -4; 0; 1], u2 = [7; -7; -4; 1]
Find an orthogonal basis for W by performing the Gram Schmidt proces to there vectors. Find a basis for W perp (W with the upside down T).
Homework Equations
Gram...
First off, I've never taken a differential equations class. This is for my Math Methods for Physicists class, and we are on the topic of DE. Unfortunately, we didn't cover this much, so most of what I am about to show you comes from the professor giving me tips and my own common sense. I'd...
Homework Statement
Let s be the linear transformation
s: P2→ R^3 ( P2 is polynomial of degree 2 or less)
a+bx→(a,b,a+b)
find the matrix of s and the matrix of tos with respect to the standard basis for the domain
P2 and the standard basis for the codomain R^3
The Attempt at a...
Homework Statement
I'm trying to understand the preferred basis problem in the foundations of QM
Ok so I read somewhere that in general any state can be decomposed in different ways.
I don't quite see how this is meant to work
Suppose 'up' / 'down' represent z component of ang mom...
Homework Statement
This is the question on my assignment:
In each case below, given a vector space V , find a basis B for V containing the linearly independent set S ⊂ B.
It has a bunch of different cases but I think that if you help me with the following two, I will learn enough to do...
Homework Statement
The Attempt at a Solution
So first I thought to myself that the proper way of doing this problem was to construct each of the standard basis vectors as a linear combination of the basis given us. I have,
T(1,0,0) = \frac{1}{2} T(1,0,1) + \frac{1}{2} T(1,0,-1) =...
Homework Statement
Let V = P_n(\textbf{F}) . Prove the differential operator D is nilpotent and find a Jordan basis.
Homework Equations
D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1}
The Attempt at a Solution
I already did the proof of D being nilpotent, which was easy. But we haven't covered...
I have S= {(1,1,0,1) (1,0,-1,0) (1,1,0,2)} its one of the subset and second it T=
{(x,y,z,2x-y+3z)}
If you were to use Gram-Schmidt method to find the orthogoan basis for T who would you processed?
I really don't understand this concept.
I know from T , the hyperplane is 2x-y+3z so the...
Does anyone know if a set of homogeneous polynomials forms a reduced Grobner basis, then they form a regular sequence in the polynomial ring? Any references?
All the references that I have looked at (so far) have not related the two.
If this is not true, can you give me a counterexample...
Hi, I have some very basic questions regarding electron energy levels/states.
In the basic atom model when an electron becomes excited (i.e. absorbs a photon or collides with a nearby atom or particle) and moves into an energy state greater than its ground state, must it always eventually...
Homework Statement
The Attempt at a SolutionSince A is a vector in V and since the A_i form a basis, we can write A as a linear combination of the A_i. We write A = x_1 A_1 + ... + x_n A_n. Thus, we have,
<x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i>. Because...
Hello Forum,
a lattice is a set of points. We can place a basis at each set of points.
The basis can be one atom or a group of atoms.
I thought that a translation of the basis would produce the whole crystal...
How is a basis different from the unit cell? Are they the same thing...
Background:(you might not be interested so you can skip if you want)
I am trying to learn general relativity using the Book Gravitation by Misner, Thorn and Wheeler. The book for the most part seems easy for me to understand but once in a while words i neither heared nor can find the meaning of...
Hello everyone,
I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise.
Specifically, I'm having a hard time understanding the matrix elements of the density matrix...
Hi,
I'm working on an example question with the following info:
\alpha = {(3,0,1) , (3,1,1), (2,1,1)} \beta = {(1,1), (1,-1)} Are a set of bases. [T]\beta\alpha = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say:
Let T: R3--> R3 be the transformation whose...
Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as
gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi
http://en.wikipedia.org/wiki/Curvilinear_coordinates
Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian...