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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.

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Change of Basis - Help

Trying to figure out how to change matrix's basis. Keep getting the opposite change of basis matrix... (Ex4.7.7): $B =([1,2]T, [3,4]T), C = ([7,3]T, [4,2]T)$ Find $P_{B<-C}$ and $P_{C<-B}$ I ...
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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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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 $\mathbb{R}^2$...
OftenConfused's user avatar
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1 answer
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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 ( {...
David's user avatar
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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 $Ax = y$, $Tx = x'$, $A'x' = y'$, $T^{-1}y'=y$ $...
Егор Лебедев's user avatar
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1 answer
21 views

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 = \...
David's user avatar
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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 ...
JerSci's user avatar
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6 votes
4 answers
566 views

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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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 ...
asdfgh jkl's user avatar
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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\...
Barto_Wynne12's user avatar
9 votes
2 answers
736 views

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 ...
Sam's user avatar
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1 answer
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about change of basis matrix and matix similarity

so lets say A,B are similar matrices and P is a change of basis matrix from A to B. so by matrix similarity $A = PAP^{-1}$ is the order of P and P^-1 important? ($P^{-1}AP = PAP^{-1}$) if yes what is ...
tensai's user avatar
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Converting between diagonalizations of a linear transformation with respect to different basis

I was originally asked to find a transition matrix $T$ and a diagonal matrix $D$ such that $D=T^{-1}[\theta]_{e,e}T$ for matrix representation of a linear transformation $\theta:\mathbb R^3 \to \...
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1 answer
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Rotation along line in $\mathbb R^3$

Let $\rho: \mathbb R^3 \to \mathbb R^3$ denote the linear transformation that is a $\pi/4$ anticlockwise rotation around the line generated by $\begin{bmatrix} 1 \\ 1 \\ 0\end{bmatrix}$. i.e., $\rho\...
Jason Xu's user avatar
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On the change of basis matrix.

We know that if you have a vector $x$ and two bases $B$ and $B'$ of a vector space $V$ then $$x_{B}=B^{-1}B'\cdot x_{B'}$$ so now let's assume we have a linear mapping $f:V\to W$ and two bases $C,C'$ ...
Amazing's user avatar
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1 answer
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Finding a basis for a linear transformation where the basis is not given

What would be the right way to approach the following question? Consider the linear transformation $\theta: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $\theta(x) = Ax$ where $A = \begin{bmatrix}...
michaeltodd's user avatar
1 vote
1 answer
61 views

Rotation matrices and change of basis

brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation ...
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Question regarding bases of bilinear forms

Let a bilinear form $\beta \in \mathcal{B}(\mathbb{R}^{(2n+1)})$, $n\in\mathbb{N}$, and suppose $detH(\beta, A) \neq 0$. Does there exist a basis $A'$ such that $H(\beta, A) = -H(\beta, A’)$ Recall ...
Avgustine's user avatar
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1 answer
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Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.

I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination': Short ...
Newbie1000's user avatar
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0 answers
23 views

Effect of Basis Change on Absolute Vector Magnitudes

Does Basis Change Affect the Absolute Magnitude of Vectors? How does a change in basis impact the absolute length of vectors? I'm trying to understand the effect of a basis change on the absolute ...
fatFeather's user avatar
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1 answer
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Finding change of basis matrix for bilinear form

I'm having a hard time finding change of basis matrix for bilinear form. I'm given a matrix $A = \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 &-1\\ 1 & -1 & 1 \\ \end{bmatrix}$ and I ...
Avgustine's user avatar
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0 answers
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Link between matrix columns and change of basis

In doing some analysis in finite dimensional space, the following question arose about the link between columns of a matrix under different basis. More precisely, consider $(e_i)$ the canonical basis ...
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Existence of an orthogonal basis with respect to a given set of matrices

Consider a complex vector subspace $W \subset V$ such that $\mathrm{dim}W = n$. Suppose I have a set $S$ of $n$ linearly independent complex matrices $S = \{ M_1, \cdots, M_n\}$ that act on $V$. Let $...
Eric Kubischta's user avatar
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27 views

Change-of-basis without using orthogonal as new basis

I have a question in my exam which is: Let :v = [3,1] and T : R2 -> R2 denote the transformation such that T(x) is the vector in span{v} nearest to x Let :0<a<pi/2 denote the angle between ...
Scotland Wind's user avatar
1 vote
1 answer
67 views

A domain-covariant notation for functions?

Note: I'm using the terms "covariant" and "contravariant" a bit loosely in this question. The standard function notation seems to be naturally codomain-covariant and domain-...
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33 views

Where am I getting lost in basis change.

For a linear transformation F we're given that F(0,1,0)=(1,1,0), F(0,0,1)=(1,-1,2), and that (1,1,1) is an eigenvector with eigenvalue=1. Find the transformation matrix for F. I know this can be ...
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Matrix Exponential and Conjugation of Jordan Normal Form and Real Normal Form

Given a diagonalizable matrix $A \in \mathbb{R^{N \times N}}$, we can decompose $A$ in following terms: $A = V \Lambda_{C} V^{-1}$ where $V, \Lambda_{C} \in \mathbb{C^{N \times N}}$ $A = Q \Lambda_{R}...
lostintimespace's user avatar
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Do basis vectors which are represented with respect to their own basis become the standard basis vectors?

Whilst studying linear algebra I came accross an at first sight peculiar conclusion. The double basis representation of the Identity linear transformation equals the Identity matrix of same dimension ...
Geralt's user avatar
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1 vote
1 answer
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Representing Complex Vector Spaces as Real Vector Spaces

Suppose $V$ is a complex vector space with respect to basis $v_{1},...,v_{n}$, and $T: V \rightarrow V$ is a linear transformation with matrix representation $A$. Now, consider $V$ as a real vector ...
pseudobulbose's user avatar
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2 answers
85 views

Change of Basis over $\mathbb{Z}$

$[v ]_B$ coordinate vector of v with respect to $B$ $[v ]_C$ coordinate vector of v with respect to $C$ ${}_C[ Id_V ]_B$ change of basis matrix from $B$ to $C$ For example we have the following set ...
False Equivalence's user avatar
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1 answer
54 views

trouble with a change of basis

I have two O.N sets $\{|e_i\rangle\}_{i=1}^r$ and $\{|\tilde e_i\rangle\}_{i=1}^r$ Then there is gotta be a change of basis matrix C, such that $|\tilde e_i\rangle = \sum_{j=1}^rc_{j,i}|e_j\rangle$ I ...
some_math_guy's user avatar
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1 answer
53 views

Change of Basis in Game Development

I recently went back to re-learning some bits of linear algebra: change of basis. And, as an exercise, I decided to revisit a video game mechanic: portals. As this youtuber puts it, the math behind it ...
Daniel Marques's user avatar
1 vote
0 answers
25 views

Theoretical question about finding the basis of an image.

Let's say that you calculate the basis of the kernel and it spans one vector. Then let's assume the rank of the matrix is two. Normally, you would choose the columns corresponding to the pivot points ...
Newbie1000's user avatar
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0 answers
29 views

Is PCA invariant up to a change of basis?

Let $X=(x_1,x_2,...,x_n)$ where $x_i \in \mathbb{R}^d$ with $d>=2$. And where each $x_i = \sum_{j=0}^{d}a_j\cdot e_j$ where $(e_1,...,e_d)$ forms a basis in $\mathbb{R}^d$. We define the first ...
Tumirsito's user avatar
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1 answer
41 views

Transition matrices in $\mathbb{R}^n$: how to compute

I was reading "Elementary Linear Algebra Applications" by Howard Anton, and in section 4.6, Change of Basis, it talks about finding the transition matrix, $P$, from an old basis $B$ to to ...
SupersonicMan12's user avatar
2 votes
0 answers
30 views

Change of basis to a nonnegative matrix

Let $A$ be an arbitrary symmetric, square matrix. I would like to use the Perron-Frobenius theorem, but I cannot do that directly since $A$ is not necessarily a non-negative matrix. So I had the ...
Fernando's user avatar
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1 answer
34 views

Find a basis for which the matrix of the following bilinear form has the form described in Sylvester's theorem

Let V be the vector space of real valued polynomials with degree smaller equal 3 and let $$(p,q) = \int_{-1}^{1} p(x)q(x) \,dx - 100p(0)q(0)$$ Is the bilinear form degenerate and find a basis for V in ...
Fregheit Meier's user avatar
1 vote
2 answers
61 views

Rotating the axes and changing variables

Without using matrix inversion, what is the easiest way to show that, for variables $u$ and $v$ corresponding to rotating the $x-y$ axes by $\theta$, we have $$ x=u\cos \theta -v \sin \theta\\ y=u\sin\...
sam wolfe's user avatar
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Why are $PAP^{-1}$ and $P^{-1}AP$ equal when changing base?

I have a doubt about performing changes of basis in linear algebra. I know that for each pair of bases, say $B$ and $C$, I can find a certain "translation" matrix $P$ that sends my vectors ...
Luca Seggiani's user avatar
1 vote
1 answer
58 views

Change of basis of a matrix - what am i doing wrong?

I want to change the basis from: $$(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$$ to $$(|00\rangle,|u_2\rangle,|u_3\rangle,|11\rangle)$$ , where $|u_2\rangle, |u_3\rangle = \frac{1}{\sqrt{2}}(|01\...
Kobamschitzo's user avatar
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0 answers
22 views

Solving for a conjugation matrix from observed transformations

I am trying to find the matrix $M$ satisfying: $$v_t'= MA_tM^{-1}v_t$$ For a dataset of observed transformations $(v_t',A_t,v_t)_t$. Basically I have two isomorphic vector spaces $U$ and $V$, where ...
hamza keurti's user avatar
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0 answers
61 views

Understanding coordinate vectors

Let's consider a vector space V with $\dim(V)=3$. By taking the basis S = ($\vec{i}, \vec{j}, \vec{k}$) with $\vec{i}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ , $\vec{j} = \begin{bmatrix}0\\1\\0\end{...
posfn0319's user avatar
1 vote
1 answer
63 views

Algorithm to put find a similar matrix under reduced form

I would like to prove that the following algorithm to find a similar matrix under reduced form, that is a triangular one (it includes Jordan normal form), works but there is one step putting me in ...
G2MWF's user avatar
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4 votes
1 answer
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How would understanding 'change of basis' explain where I went wrong?

(Sorry in advance if I don't get all the terminology right, and for the lengthy question!) Intro I've got two $2\times 2$ matrices called $\textbf{M}$ and $\textbf{N}$, which represent the linear ...
Anis Manuchehri-Ramirez's user avatar
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Determining Change of Basis Matrix for Linear Transformation from Eigenbasis to Standard Basis

I am seeking assistance in understanding and computing the change of basis matrix (M) for a linear transformation from the Eigenbasis (B) to an Standard Basis (A). Specifically, I have a linear ...
Rishav Dhariwal's user avatar
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0 answers
54 views

Change of basis and transformation of a vector.

I am given the matrix $A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$, and a vector $x=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$ After computing the eigenvalues and normalized eigenvectors, we ...
Akis's user avatar
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-2 votes
1 answer
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Question Regarding a Proof on Change of Basis [closed]

Let $v_1,v_2,v_3\in \mathbb {R}^3$ be basis vectors. Suppose there exists $3 \times 3$ matrix A and distinct numbers $\lambda_1,\lambda_2,\lambda_3$. Then $Av_i = \lambda_iv_i$ for $i = 1, 2, 3$. And $...
Avgustine's user avatar
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1 answer
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What is $P$ in Theorem 2.4.8 Linear Algebra by Hoffman and Kunze?

Code borrowed from here Suppose $P$ is an $n\times n$ invertible matrix over $F.$ Let $V$ be an $n$-dimensional vector space over $F,$ and let $\cal B$ be an ordered basis of $V.$ Then there is a ...
pie's user avatar
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0 answers
55 views

About algebras such that $ j_a j_b \in \{ -1,0, +1,j_c,-j_c \}$ for all $a,b$

Consider a commutative unital algebra $A$ of finite dimension $n>3$ over the reals. The product is defined such that elements are generated with real number coefficients $(a_0, \dotsc, a_n) $ for ...
mick's user avatar
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2 votes
1 answer
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How is $a_j'= \sum\limits_{i=1}^n P_{ij}a_i$ shouldn't it be $a_i = \sum\limits_{j=1}^n P_{ij}a_j'$? Hoffman and Kunze theorem 2.4.8

How is $a_j'= \sum\limits_{i=1}^n P_{ij}a_i$ shouldn't it be $a_i = \sum\limits_{j=1}^n P_{ij}a_j'$ I tried my best to understand how this is true and I spent more than an hour here another question ...
Mathematics enjoyer's user avatar

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