Questions tagged [change-of-basis]
This tag is for question about changing basis of a finite dimensional vector space. For example, how does the representation of a vector, or a matrix change with the change of basis. Please don't use this tag on its own, it is better to add a more general tag which is relevant to your question, e.g. [linear-algebra] or [matrices] for better visibility.
1,190
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Is the gradient the equivalence class of all spanning vector bases of the tangent vector space at a manifold point? [closed]
When I try to spell out what this means the discussion becomes complicated and verbose. So I will simply ask. Is it correct to say that in finite dimensional real number differential geometry the ...
- 3,349
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1
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coordinates of the vector relative to the new basis
Given vectors $(v, b_1, b_2, b_3,...,b_n)$ defined by their coordinates in an arbitrary basis. Prove that the vectors $(b_1, b_2, b_3, \ldots, b_n)$ form a basis and find the coordinates of the vector ...
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Understanding Equivalence of Matrix Elements in Different Bases for Hermitian Operators
Suppose $Q$ and $R$ are two system (which are represented by state vectors in the vector space V) on the same vector space $V$
$|i\rangle$ is an ortonormal base of $V$
$|i_R\rangle$ is an ortonormal ...
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0
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1
answer
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Question about the change of basis matrix
I want to ask a question to all of you to see if I can clarify this concept. If I have two basis, $b_1$ and $b_2$ that are not the canonical basis, i can find the change of basis matrix from $b_1$ to $...
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1
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Order in the isomorphism between $l_2 \times l_2$ and $l_2$
I know that $l_2 \times l_2$ is a Hilbert space and so it must be isomorphic to $l_2$, however I'm looking for how exactly does this isomorphism works because the basis of $l_2 \times l_2$ is ...
- 800
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13
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Calculate Components of square integrable functions w.r.t. some basis
Consider the space of square integrable functions on the non negative real numbers $L^2(\mathbb{R}_0^+)$. I found out, that the Laguerre functions modulo some normalization define an orthonormal basis ...
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1
answer
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Inconsistent result using matrix for non-standard basis
I am making what I suspect is a very basic error and would like to know where I"m going wrong. In short, I am developing a matrix for a linear mapping using a non-standard basis for $\mathbb{R}^2$...
1
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1
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How to calculate the matrix representation of a quadratic form?
I'm working on a problem involving quadratic forms and I need some help verifying my calculations. Here is the problem:
I'm given a quadratic form ( q(x, y, z) = x^2 + 2yz ).
The basis ( B ) for ( {...
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1
vote
1
answer
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Why formula for operator in another basis is like this?
We have operator $A$ in basis $E$. Transformation matrix from $E$ to $E'$ is $T$. There is a formula for $A$ in new basis $E'$ : $A' = T^{-1}AT$.
We got $Ax = y$, $Tx = x'$, $A'x' = y'$, $T^{-1}y'=y$
$...
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1
answer
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How to find the representing matrix of a quadratic form with respect to a non-standard basis?
Given the quadratic form $( q(x, y, z) = x^2 + 2xy + 6xz + 4yz + z^2 )$, I know how to find the representing matrix for the standard basis. The representing matrix for the standard basis is:
$$
Q = \...
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votes
1
answer
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How to find the basis vector of a transformed frame?
I have a frame B that is rotated w.r.t to frame A about the z axis by 30 degrees clockwise and translated by [2, 0, 0]. Frame A is translated by [1, 0, 0] w.r.t to the world frame. The goal is to ...
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4
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Dual space isomorphism non-canonical choice example
In a lot of resources that I have read it is mentioned that the isomorphism between $V$ and $V^*$ is non-canonical, but I was never sure that I properly understood precisely what this means. I haven't ...
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0
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Calculate the Basis B and C
Let
$
A=\left[\begin{array}{cccc}
2 & 3 & 2 & 3 \\
3 & 4 & -1 & 1 \\
1 & 1 & -3 & -2
\end{array}\right]
$
and $ f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} $ the ...
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1
vote
0
answers
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Jacobian change of basis for integration
I just wanted to see if anyone could verify my work.
I'm trying to integrate the function $f(x,y) = xy$ over the circle of center $(1, 2)$ and radius $2$. The conversion I am using is:
$x = r\cos\...
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9
votes
2
answers
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What does it mean to say that a linear transformation *is* the change of basis matrix?
I wish to check my understanding on part of the proof of Proposition 5.3 in Lee's Introduction to Smooth Manifold. It reads as follows: $\def\tE {\widetilde{E}}$
Let $(E_i)$ and $(\tE_i)$ be two ...
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