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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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1answer
38 views

Find basis for $\Bbb Q(\sqrt 2 + \sqrt[3]2)$ over $\Bbb Q(\sqrt 2)$ [on hold]

I need to find the basis for this which is apparently $\{\sqrt[3] 2, (\sqrt[3] 2)^2, 1\}$. Thanks so much for your help
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46 views

How many bases can be formed given a set of vectors? [duplicate]

Let $v_1,\cdots,v_n$ be a set of vectors such that the generated vector space has dimension $r$. How many different bases with dimension $r$ can be formed from these vectors? Don't understand the ...
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36 views
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2answers
28 views

Understanding diagonalization through change of basis

I'm sorry if what I am asking is unclear, but I was hoping that somebody might be able to help me understand diagonalization a bit more by explaining it through the lens of a change of basis. I ...
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1answer
11 views

Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? $P$ is the change of basis matrix from $B$ to $C$ and $T$ is a linear transformation

Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? ($P$ is the change of basis matrix from $B$ to $C$ (both vector spaces) and $T$ is a linear transformation). It would make a lot of ...
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1answer
24 views

Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B = $ {$1+x, 1-x, x^2$}

Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B = $ {$1+x, 1-x, x^2$} To find the matix of a transformation with respect to a given basis, I find the images of the basis ...
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0answers
16 views

Basis functions and weak ODE solution

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = \lambda u$. Given basis functions, call them $\phi_i$, I use a variational procedure and the Ritz method ...
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3answers
39 views

How do you convert $6.75$, in base $10$, to base $2$?

Can I get a detailed explanation on how to convert $6.75$ base $10$ to $2$. I have really tried all I can but I don't still get it.
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1answer
17 views

Why does row reducing an augmented matrix of two basis written in the coordinates of a third give the change of basis matrix?

Sorry for the longwinded and vague title. I've included a full picture of the theory (taken from my phenomenal textbook, David Poole's Linear Algebra: A Modern Introduction-Brooks Cole (2014)) below. ...
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1answer
53 views

Change of basis of the kernel of a rectangular matrix

Suppose that I have a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $T$ can be written as an $m \times n$ matrix. Let $k = \dim\ker T$, where $k > 0$. Thus, $\ker T \...
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0answers
40 views

Is inner product preserved on change of basis?

Suppose we have a matrix $A$ over which an inner product is defined. Let us denote this inner product by $\langle, \rangle_A$. Now let us suppose that this matrix $A$ is symmetric. Then there exists ...
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1answer
38 views

A difficulty in understanding an example from section 3 Vinberg.

This example was on p. 38 in Vinberg, and it was on extension of the ground field (where here the ground field is $\mathbb{R},$ and we will extend it to $\mathbb{C}$): But I have a difficulty why the ...
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0answers
7 views

Transition-Matrix for perspectives of two stereo cameras which capture two pointclouds of the same object

I wrote code which i want to transform the pointcloud of one of the cameras coordinatesystem into the other.I know three Points(xyz) from each perspective: (x1,y1,z1) and (x1',y1',z1') for all three ...
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0answers
14 views

Vieta's Formula for Chebyshev basis

Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
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1answer
64 views

Matrix of endomorphism in different basis

Let $E$ be a vector space, with $\dim E = n$. Let $u$ be an endomorphism of $E$ and $B = (b_1, \dots, b_n)$, $C = (c_1, \dots, c_n)$ be two basis of $E$. Let $\operatorname{Mat}_{BC}(u)$ denote the ...
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0answers
15 views

How can I get transformed coordinate between different basis?

For example, there is set of DCT basis which are orthogonal: $F_1(x,y),F_2(x,y)\cdots,F_N(x,y)$. So, given function can be uniquely expressed as sum of DCT basis. (we decompose an image to sum of DCT ...
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1answer
22 views

Matrix representation of a linear transformation under change of basis.

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$ such that $x\mapsto Ax$. Suppose A is diagonalizable and $B=\{v_1,...,v_n\}$ is a ...
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1answer
44 views

Change of basis…

**Question:**Let $(1, z, z^2)$ be the standard basis for $\mathbb{C}^2$. Let $(1, z - 1, (z - 1)^2$ be another basis for $\mathbb{C}^2$. Find a matrix transformation from $p(z) = a_0 + a_1z + a_2z^2$ ...
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1answer
12 views

Question regarding possible implicit change of basis when performing 2D affine translation

Lets say we have a point that is a part of 3d space, but fixed to the $z=1$ plane. That is, we have, in column matrix form, assuming the canonical $\mathbb R^3$ basis: $$ v_0 = \begin{bmatrix} ...
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2answers
68 views

What would a change of basis matrix look like between the abstract vector spaces of $\cos^k(x)$ and $\cos(kx)$?

My friend mentioned that creating this matrix is possible and useful for difficult integrals with high power cosine functions. However, I am having trouble actually creating it. My thoughts are to ...
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1answer
22 views

Linear map $M_{matrix} \mapsto v_{vector}$

In my algebra's workbook there is this exercise that I don't know how to approach ... so far I have only dealt with linear maps of the type $vector \mapsto vector $, I've never seen $matrix \mapsto ...
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1answer
41 views

Let $ w_1,w_2,\dots,w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$

Let $w_1, w_2,\dots, w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$ How do I find the scalars $a_1, a_2$, and $a_n$ ?
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59 views

Matrix of a linear mapping in the new basis

Let $N, M$ be a linear spaces over the same field $\mathbb{F}$. So $\{e_i\}$, $\{\overline{e_i}\}$ -- two bases in $N$ and $\{e_k'\}$, $\{\overline{e_k}'\}$ -- two bases in $M$. Let $f$ be a map ...
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0answers
36 views

Basis vectors in different coordinates

Consider the point P(1,1,1). In cylindrical polar coordinates, this can be expressed as $\sqrt(2)e_r$ + $e_z$ . I am unsure as to why it cannot be expressed as $\sqrt(2)e_r$ +$\frac{\pi}{4}$ $e_{\...
3
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1answer
32 views

How to compute the Jacobian matrix of a multivariate function in a nonstandard matrix?

Given a function $f:R^2\rightarrow R^2$ such that $f(x,y)=(xy, \cos xy)$, I need to compute the Jacobian matrix Df with respect to the basis $\{(1,0), (1,1)\}$. Not confident in my answer though. ...
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1answer
21 views

Why can't we find a standard matrix representation in terms of eigenvalues with only ones on the diagonal?

For every linear map $M: V \to W$ between real finite dimensional vector spaces one can always choose a basis of $W$ and one of $V$ so that the matrix representation of $M$ has $n :=$rank$(M)$ ones ...
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2answers
24 views

transition from one base to another

Given the polynomials $p_1,p_2,p_3,v_1,v_2,v_3 \in P_2$: \begin{gather*} p_1(t) := t^2 − 2t + 5, \qquad p_2(t) := 2t^2 − 3t, \qquad p_3(t) := t + 1, \\ v_1(t) := t^2 + 4t − 3, \qquad v_2(t) := t − 1,...
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0answers
23 views

An affine transformation, and its effect on a curve and a polynomial.

Suppose $(u,v) = A(x,y)$ is affine transformation. Where $u = ax + by + e$, and $v = cx + dy + f$ , and the inverse transformation given by $x = a'u + b'v + e'$ and $y = c'u + d'v + f'$. Suppose ...
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1answer
23 views

changes of coordinates (effect on points and curves)

One thing I am finding challenging in my studies is in a situation involving two coordinate systems - the $x,y$ and $u,v$ of a real plane. My understanding is that when relations such as the ...
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1answer
24 views

changes of coordinates, confusing book language.

A book i'm reading says consider the "dictionary" $x = u + a$ and $y = v + b$ between the $x,y$ and $u,v$ coordinate systems, where the $u,v$ coordinate axes are obtained by translating the $x,y$ $a$...
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1answer
60 views

Matrix representation of the linear operator $A = \frac{d}{dx} + x \frac{d}{dx}$

$A = \frac{d}{dx} + x \frac{d}{dx}$ is an operator that acts on the vector space $P^n$ of all real valued polynomials of degree $\le n$. with respect to the standard basis $\{x_n\}_{n\in \mathbb{N}_0}$...
3
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1answer
56 views

Laplace transform and frames vs Bases

The Laplace transform $$F(s) = \int^{∞}_{0}f(t)e^{-st} dt$$ can be understood much like the fourier transform, as a change of basis of an $L^2$ function to the eigen functions of the differential ...
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1answer
21 views

Three bases, one linear map, find vector $x$

Given three bases $B, C$ and $D$ and linear map $f:\Bbb R^2\to\Bbb R^2$, and $x$ from $\Bbb R^2$. We also know that $[x]_B=(x_1,x_2)^T$. $$[f]_{B\to C}=\begin{bmatrix}2 & 3 \\3 & 1 \end{...
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1answer
30 views

What is the general method of expressing a basis in terms of another basis of a vector space?

I want to understand basis of vector spaces more clearly. I know that the basis of a vector space is a set of linearly independent elements that span that vector space. I know that an element in a ...
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3answers
37 views

Change of base in $R^2$

In the official solution of an university exam I see the following question: Given the base vectors $$ B = \{ \begin{pmatrix} -1 \\ 1 \end{pmatrix} , \begin{pmatrix} 1 \\ 1 \end{pmatrix} \} $$ and ...
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3answers
25 views

Finding linear transformation from matrix

I tried solving this question by taking given matrix $[A]_{B_{1}}$ = $[T]_{B_{2}}$. The answer that I am getting is $(35,-10)$. Tried solving several times and still reaching at the same answer. ...
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0answers
54 views

Prove that there exists a semi-orthogonal $U$ such that $U^TAU=B$, where $A$ and $B$ are positive-definite symmetric matrices.

Let there be a semi-orthogonal matrix $U \in \mathbb{R}^{m\times n}$ such that $U^TU=I_n$ if $m > n$ If $A \in \mathbb{R}^{m\times m}$ and $B \in \mathbb{R}^{n\times n}$ are positive-definite ...
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1answer
65 views

Show that the associated matrix is $PAP^{-1}$

Let $\eta \in \operatorname{End}_{R}(R^n)$ and let $A$ be the associated matrix of $\eta$ with respect to the basis $(e_1, \dots e_n)$. Let $f_i = \sum p_{i,j}e_j$ where the matrix $P=(p_{i,j}) \in ...
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0answers
25 views

Transformation matrix with 2 bases

I have the following: $ C = ($$ \left[ \begin{array}{cc} 1\\ -1 \end{array} \right], \left[ \begin{array} -1\\ 2 \end{array} \right]) , B = ($$ \left[ \begin{array}{cc} 2\\ 1 \end{array} ...
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1answer
47 views

Is a Change of Basis Matrix equivalent to the matrix inverse in this case?

I was looking at how to construct a change of basis so that when given a system of linear equations one could change the associated matrix into a diagonal matrix -- thus making the system easier to ...
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2answers
57 views

Intuition for the invariance of the determinant under change of basis

$$A' = PAP^{-1}$$ $$\det(A')=\det(P)\det(A)\det(P^{-1})=\det(A)$$ Now, that makes sense algebraically, but consider the below diagram: This a geometric representation of the two 'normal' basis ...
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2answers
34 views

How to find a matrix that transforms vectors from basis $B$ into vectors from basis $C$? [closed]

$$B=\left\{\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\right\} \quad C=\left\{\begin{bmatrix} 4 \\ 5 \\ 6 \end{...
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1answer
29 views

Let $(v_1, v_2)$ be a basis for $\text{Im}(p)$ and $(v_3)$ be a basis for $\ker(p)$. Prove that $B' = (v_1, v_2, v_3)$ is a basis for $V$ .

Let $B = (1, X, X^2)$ be a basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$...
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2answers
20 views

How to do the following change of Basis

$V=\mathbb{Q}^3$ The two following Bases of V are given $S=\{(1,0,0) , (0,1,0) , (0,0,1)\}, T=\{(-4,2,1) , (-1,0,1), (1,-1,1)\}$; Let $w ∈ V$ with $γ_T$$(w)$ =$ $$\begin{pmatrix}1\\0\\1\end{pmatrix}$...
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1answer
18 views

Two Change of Basis Matrices

Let $V$ $=$ $P_2$ and $dim(v)$ $=$ $3$. Let $B$ $=$ {$1$, $x$, $x^2$} and $S$ $=$ {$1+x$, $2-x$, $3+x^2$} be two basis of $V$. What is the matrix $P_B,_S$ and $P_S,_B$? I am not entirely sure how to ...
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1answer
21 views

Compute $\text{rk}(p)$ using Gauss reduction on $A$. Compute $\dim\big(\ker(p)\big)$.

Let $B = (1, X, X^2)$ be a basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$...
0
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1answer
43 views

Prove that $p ◦ p = p$. (Representing a Linear Transformation as a Matrix)

Let $B = (1, X, X^2)$ be an ordered basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}(\mathbb{R}_2[X])$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)...
0
votes
2answers
35 views

Transformation matrix with respect to an orthonormal basis

I have this question here... Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$. $(a)$Construct an orthonormal basis $B'$ for $V$ (usual dot product). $(b)$ ...
0
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0answers
69 views

Christoffel symbol transformation

The Christoffel symbols transform like $$\Gamma^{\prime a}_{bc} = \frac{\partial x^{\prime a}}{\partial x^d} \frac{\partial x^e}{\partial x^{\prime b}} \frac{\partial x^f}{\partial x^{\prime c}} \...
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2answers
70 views

Finding B-spline for space spanned by Multi-dimensional Spline.

It's known that a B-spline of degree $p$ , $B_j^p(x)$ is completely determined by a knot vector $(u_j,u_{j+1},...,u_{j+p+1})$. One could define it as: $B_j^p(x)=[u_j,u_{j+1},...,u_{j+p+1}](\cdot - x)^...