What is Change of basis: Definition and 97 Discussions
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.Such a conversion results from the change-of-basis formula which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written
x
o
l
d
=
A
x
n
e
w
,
{\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },}
where "old" and "new" refer respectively to the firstly defined basis and the other basis,
x
o
l
d
{\displaystyle \mathbf {x} _{\mathrm {old} }}
and
x
n
e
w
{\displaystyle \mathbf {x} _{\mathrm {new} }}
are the column vectors of the coordinates of the same vector on the two bases, and
A
{\displaystyle A}
is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
I am looking at the point group C<sub>3v</sub> described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form.
What is the rationale for choosing the new basis. Is it...
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.
For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.
The expressions have similar form except for the order of the inverses. Is there there any connection...
Until now in my studies - matrices were indexed like ##M_{ij}##, where ##i## represents row number and ##j## is the column number. But now I'm studying vectors, dual vectors, contra- and co-variance, change of basis matrices, tensors, etc. - and things are a bit trickier.
Let's say I choose to...
If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|##
##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|##
And we've a general vector ##|\alpha\rangle## such that...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
Hello,
Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There...
Hello,
I am studying change of basis in linear algebra and I have trouble figuring what my result should look like.
From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...
I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts.
It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
Let
## \begin{align}M =\begin{pmatrix} 2& -3& 0 \\ 3& -4& 0 \\ -2& 2& 1 \end{pmatrix} \end{align}. ##
Here is how I think the JCF is found.
STEP 1: Find the characteristic polynomial
It's ## \chi(\lambda) = (\lambda + 1)^3 ##
STEP 2: Make an AMGM table and write an integer partition...
Anyone know how to change a basis of a qubit state of bloch sphere given a general qubit state? There are 3 different basis corresponding to each direction x,y,z where |1> ,|0> is the z basis, |+>, |-> is the x basis and another 2 ket notation for y basis.
Given a single state in the x basis...
Homework Statement
Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...
I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle:
\begin{bmatrix}
E_0 &-A \\
-A & E_0
\end{bmatrix}
Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix}
\frac{1}{\sqrt{2}}\\...
I think I do not quite understand the role that change of basis plays is superpositioning of states.
If there is an observable, ##A## which is represented by the operator ##\hat{A}##, then the set of observed values for that observable will be the set of eigenvalues defined by the operator...
I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the...
Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A.
Now, the first (old) basis, in terms of the second(new) basis, is given by, say,
$$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$
How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for...
Hello everybody,
My question is about change of frame of reference and its consequences on an ordinary function.
Let's ##B## a frame of reference that is linked to another one ##B'## through a change of basis matrix ##M##.
So, for an equation written in the first basis ##B## as...
I have two n-vectors e_1, e_2 which span a 2D subspace of R^n:
V = span\{e_1,e_2\}
The vectors e_1,e_2 are not necessarily orthogonal (but they are not parallel so we know its a 2D and not a 1D subspace). Now I also have a linear map:
f: V \rightarrow W \\
f(v) = A v
where A is a given n...
Dear all,
The Hamiltonian for a particle in a magnetic field can be written as
$$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$
where ##\boldsymbol\sigma## are the Pauli matrices.
This Hamiltonian is written in the basis of the eigenstates of ##\sigma_z##, but how is it...
This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##).
In my more...
Consider the following set of vectors in $\mathbb{R}^3:$ $u_0 = (1,2,0),~ u_1 = (1,2,1), ~u_2 = (2,3,0), ~u_3 = (4,6,1)$ Explain why each of the two subsets $B_0 = \left\{u_0, u_2,u_3\right\}$ and $B_1 = \left\{u_1, u_2, u_3\right\}$ forms a basis of $\mathbb{R}^3$. If we write $[\mathbf{x}]_0$...
This is Exercise 2.20 in Nielsen and Chuang's Quantum Computation and Quantum Information, on page 71.
Suppose $A'$ and $A''$ are matrix representations of an operator $A$ on a vector space $V$ with respect to two different orthonormal bases, $|v_i\rangle$ and $|w_i\rangle$. Then the elements...
Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##?
My steps are as follows:
##<\phi_m\,|\,A\,|\,\phi_n>##
##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr##
##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr##
By the closure...
Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague.
Basically, as I understand it, the Jacobian (intuitively) describes how surface (or volume) elements change under a...
Hi, sadly my textbook assumes a knowledge I didn't have, of change of basis matrices & coordinate systems for linear transformations; so I have been trolling around the web to fill in the gaps as best I can.
I have an open post that has no replies -...
Homework Statement
From Linear Algebra with applications 7th Edition by Keith Nicholson.
Chapter 9.2 Example 2.
Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a).
If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P such that...
I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus.
Thank you.
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...
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...
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...
Homework Statement
We have the initial orbital angular momentum state in the x basis as |l,ml>x=|1,1>x. We are asked to find the column vector in the z-basis that represents the initial orbital angular momentum of the above state. It then says "hint: use an eigenvalue equation".
Homework...
I did some linear algebra studies and learned how to change between foreign bases and the standard basis:
Change of basis matrix multiplied by the vector in coordinates with respect to the foreign basis equals the vector in coordinates with respect to the standard basis.
Of course, this is...
Homework Statement
Homework Equations
\check{T} = BTB^{-1} (eq1)
The Attempt at a Solution
Ok, so I have a couple of questions here if I may ask... First, I want to be sure I understand the wording of (a) and (b) correctly. Is the following true?:
(a)
Write the matrix T...
I want to find a matrix such that it takes a spin z ket in the z basis,
| \; S_z + >_z
and operates on it, giving me a spin z ket in the x basis,
U \; | \; S_z + >_z = | \; S_z + >_x
I would have thought that I could find this transformation operator matrix simply by using the...
Following on from a previous post of mine about linear operators, I'm trying to firm up my understanding of changing between bases for a given vector space.
For a given vector space V over some scalar field \mathbb{F}, and two basis sets \mathcal{B} = \lbrace\mathbf{e}_{i}\rbrace_{i=1,\ldots ...
Homework Statement
Hello people,
I am trying to understand a problem statement as well as the density operator, but I still don't get it, desperation is making me posting here.
The problem comes as
The problem then continues with other questions but I'm having troubles with the very first one...
I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence.
On page 419 D&F define similar matrices as follows:
They then state the following:
BUT? ... how...
Hi all,
Just doing a bit of personal study on vector spaces and wanted to clear up my understanding on the following. This is my description of what I'm trying to understand, is it along the right lines? (apologies in advance, I am a physicist, not a pure mathematician, so there are most...
Homework Statement
Let B = {(1, -2),(2, -3)} and S be the standard basis of R2
and [-8,-4;9,4]
be a linear transformation expressed in terms of the standard basis?
The Attempt at a Solution
1) What is the change of basis matrix PSB ?
1,2
-2,-3
2)What is the change of...
Homework Statement
Let A = [1 0
4 2 ]
Let B be the eigenbasis {[1,4], [0,1]}.
--Find [T]B where T(x)=A(x).
The Attempt at a Solution
Would [T]B = {[1,-1], [0,2]}?
We are trying to find [T]B, the matrix representation of T with respect to B. So would my answer...
Homework Statement
B = {b1, b2, b3}
and
C = {c1, c2, c3}
are two basis's for R3 where the connection between the basis vectors are given by
b1 = -c1 + 4c2, b2 = -c1 + c2 + c3, b3 = c2 - 2c3
a) decide the transformation matric from basis B to basis C.
A vector x is given in...
ive been revising all holidays, unfortunately I've just realized I've been finding eigenvalues using the ensemble when i may have to change basis for the exam. looks at homework questions, workshop questions... nothing!
anyway an example problem:
rewrite
|PSI> = a|0> + b|1>
in the...
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...
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}...
Homework Statement
Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...
Here is the question:
Here is a link to the question:
Matrix of change of basis? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform.
My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...
Let B={b1,b2} and C={c1,c2} be basis. Then the change of coordinate matrix P(C to B) involves the C-coordinate vectors of b1 and b2. Let
[b1]c=[x1] and [b2]c=[y1]
...[x2]...[y2].
Then by definition [c1 c2][x1]=b1 and [c1 c2][y1]=b2. I don't get how you can
....... [x2].....[y2]
multiply the...
Homework Statement
Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}.
The part that I am confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't...
I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this?
Any help really appreciated- thanks!