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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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Find basis to sub vector spaces

V = $\Bbb R_3 [x]$ and W,U $\subseteq$ V are sub vector spaces. U=$span${$1 - x, x^2, x^2-x^3, -1+x-x^2+2x^3$} W={p(x)$\in\Bbb R_3 [x]$ | p(1)=0 ^ p(2)+p(0)=0} Find basis to W, U+W, U$\cap$W
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Basis to vector space with all inverse matrices

$V$ is sub vector space of $M$(2x2) $(\Bbb R)$, V= { $\begin{matrix}a+b & a+e & \\ c-2d&4c-8d\\\end{matrix}$} when $a,b,c,d,e \in \Bbb R$. Give an example of basis to V, that ...
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Find a basis and coordinates for a second degree polynomial

Find a basis $B$ for $P_2$ $[p]_B = \begin{bmatrix}p(0)\\p(1)\\p(2)\end{bmatrix}$ and its coordinates to a second degree polynomial The solutions says: $p(x) = p(0)e_1(x) + p(1)e_2(x) + p(2)e_3(x)...
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1answer
35 views

transition matrix of dual basis

Q :Let $\left\lbrace v_i \right\rbrace^n_{i=1}$ and $\left\lbrace w_i \right\rbrace^n_{i=1}$ be basis of V and also let $\left\lbrace \phi_i \right\rbrace^n_{i=1}$ and $\left\lbrace \sigma_i \right\...
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1answer
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Scaling of basis hat functions to given domain

I would like to transform the standard function values of the hat functions from $[0,1]$ to a given domain $[-R,R]$, whereas the standard hat functions are given through: $\begin{align} \varphi(x) = \...
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the matrix of the billinear form: $ T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$

Let $B = ((1,2)^T,(1,3)^T)$ be the basis of $V=\Bbb R^2$. Find the dual basis $B^*=(e_1,e_2)$ Find the matrix of the billinear form: $ T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$ ...
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Find a Base for Map(R,R)

I have trouble finding a basis for $Map(\mathbb{R},\mathbb{R})$. While I know what a/the (standard)basis is for $\mathbb{R}^{2\times1}$, that should be $\left(\begin{array}{c} 1 \\ 0 \end{array}\right)...
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1answer
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determinant of changing of basis matrix

Let $B=\{f_{1},…,f_{n},g_{1},…g_{n}\}$ the order orthonormal basis of $2n$-dimensional and $B′=\{f_{1}+g_{1},f_{1}−g_{1},…,f_{n}+g_{n},f_{n}−g_{n}\}$ the other order basis. How can I compute ...
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change of basis $S$ to $T$

$$ S = (1,0,0), (0,1,0), (0,0,1)\\ T = (1,1,-1), (1,1,1), (1,0,1)\\ f(x,y,x) = (x+y-z , x+2y+z , z) $$ Could someone help me find the change in basis s[1]t and s[f]s For the first one i tried ...
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2answers
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Co-ordinate transformation in $ 2$ dimensions

Consider a $2$ dimensional $x-y$ co-ordinate system as given below Figure $1$, with unit vectors ${\hat{i}}$ and ${\hat{j}}$ respectively Figure 1 Now I would like to construct a different co-...
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How to find a basis of pure sinusoids

I have this little question: Imagine some set of let's say 10 sinusoids, defined on the same interval (all with different frequencies, amplitudes, phases). Then a friend gives you a linear ...
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Diagonal matrix with distinct diagonal entries is similar to a diagonal plus an lower triangular matrix

Let $A$ be non-zero matrix such that $a_{ij}=0$ $\forall i\ge j$. $D$ be a diagonal matrix with distinct diagonal entries. Now I want to show that $D$ is similar to $D+A$. Then how can I show that ...
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1answer
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Proof that if two matrices are similar then they are both matrices of the same linear transformation T but with respect to different bases

I have a question on one of the steps in the following proof. To prove: if two matrices A,B are similar then they are the matrices of the linear transformation for some matrix with respect to two ...
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2answers
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Find the transition matrix from $(1+x^2,2x,1)$ to $(1,x,x^2)$

This is a question with its solution i was wondering if it is correct and if it is or it is not can someone explain what is going on and why do we need $B=P^{-1} A P$ $B$ and $A$ are given along with ...
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1answer
20 views

Finding the matrix for a linear transformation

I have a transformation, $T(p(x)) = p(x+1) - p(x) -p'(x)$ on $R[x]_3$ (polynomials of degree 3) and I want the matrix for T. No basis is specified, so I'm assuming the standard, what should I do and ...
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2answers
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Basis and solution of System of equation

$V$ be a n-dimensional vector space over the field $F$, with fixed Basis $\{\ \alpha_1, ...\alpha_n\}$ . A system of linear equitation - $$a_{11}x_1+a_{12}x_1+a_{11}x_2 \cdots +a_{1n}x_n=0$$ $$a_{21}...
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1answer
25 views

Change of basis, where the new basis is unknown

I'm being asked a basis $B$ such that $|f|_B = \begin{bmatrix} 0&0\\ 0&1\\ \end{bmatrix} $ $ f: R^2 \rightarrow R^2 $ is linear transformation that is a projection such that $ f([x_1 x_2]^T)...
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1answer
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[Linear Algebra ]Change of basis Matrix

This is the second part of a problem I've meaning to solve, here's the First part $B = \{v_1,v_2,v_3\}\ and\\ C=\{w_1,w_2,w_3,w_4\}$ $$A= \begin{pmatrix} 1 & 1 & 2 \\ 1 & -1 &...
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1answer
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Finding a coordinate vector given an ortho-normal basis

How would I do this problem? (these are all column vectors) Find the coordinate vector $[\mathbf{v}]_b$ is $\mathbf{v}=\left[3, 5, -4\right]$ and $B$ is the orthonormal basis $$B=\{[\frac{1}{\sqrt{2}}...
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1answer
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Is this set a basis of $F_{3}^{1*3}$ or $F_{2}^{1*3}$?

Is this set a basis of $F_{3}^{1*3}$ or $F_{2}^{1*3}?$: {0,1,1),(1,0,1),(1,1,0)} This was a question on my last quiz and I decided that it was a basis for $F_{2}^{1*3}$, because there are only 1 and ...
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2answers
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On Diagonalization of a Matrix

When we try to think what a transform looks like in a coordinate system with some other basis using change of basis, we do the following, let the unknown system be A and we know about B We take the ...
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1answer
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Orthogonal bases of the vector space $\mathbb{Z}_2^4$

Let $\mathbb{Z}_2$ be the two element field $\mathbb{Z}/2\mathbb{Z}$. The vectors $e_0 = \langle1,1,1,1\rangle$, $e_1=\langle1,1,0,0\rangle$, $e_2 = \langle1,0,0,1\rangle$, $e_3 = \langle1,0,1,0\...
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2answers
24 views

Find all vectors that suit $Ax = (1, 0, 0)$ where $A$ is linear mapping

We have linear mapping give with a matrix from standard basis to basis $X$. Basis $X = ((1,0,1),(0,1,0),(0,1,1))$. The matrix looks like the following \begin{bmatrix}1&0&1\\0&1&0\\1&...
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3answers
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Prove that $a$ is a linear map and write the matrix in a given basis

Let $n \in \mathbb N$ and let $a:\,R_n[X]\to R_n[X]$ be the derivative linear map, such that $a(P)=P'$. I am aware that it must satisfy the conditions of: $f(x+y)=f(x)+f(y)$ $f(kx)=kf(x)$ However I ...
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1answer
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Homework help, how to find a base by given vectors

I need to find a base from the polynomial space $P_3[x]$ for $[1 − x + 3x^2 − x^3]_D = $$ \begin{pmatrix} 1\\ 0\\ 2\\ 0\\ \end{pmatrix}. $$ $ If it helps, in the first part ...
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1answer
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Nilpotence and module of coefficients

Given a transformation $T\in L(V)$, $T$ nilpotent where $V$ is a vector space of finite dimension, show that $\forall \epsilon >0$, there exists a basis $\mathcal B$ from $V$ such that $[T]_\...
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1answer
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Change of base matrix between displaced and rotated coordinate systems

I have a function that solves a problem when a specific angle equals $0$. The same function can be used with non-zero angles if you compute the problem from other coordinate system. The scheme of the ...
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0answers
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Given $h$ and bases $\mathcal A, \mathcal B$ , find the representation matrices $A$,$B$, and the matrix $P$ such that $B=PAP^{-1}$.

For the homomorphism $h: \mathcal P_2 \rightarrow \mathcal P_2$ given by $1 \mapsto 3$, $x \mapsto 2x-1$, $x^2 \mapsto x^2-x-1$ (i) Find the matrix $A=Rep_{\mathcal{A,A}}(h)$ for basis $\mathcal ...
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How is the Laplace Transform a Change of basis?

This question is primarily based on the following answer's way of reasoning, https://math.stackexchange.com/a/2156002/525644 If you want to write a new answer to the question; "How is the Laplace ...
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1answer
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Help a robot avoid computing the matrix inverse using the Gram-Schmidt process

Imagine you have a robot whose position is recorded as $t_1, t_2, t_3, \dots$ in its coordinate frame. Check the visualization here. We can write down the robot's basis in our 2D image plane, i.e. ...
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1answer
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Transcendence in $\mathbb{R}^2$

Let $\boldsymbol{v}_1=(x_1,y_1),\boldsymbol{v}_2=(x_2,y_2)\in\mathbb{R}^2$ be linearly independent over $\mathbb{R}$. Define $\boldsymbol{v}\in\mathbb{R}^2$ to be transcendental if at least one of ...
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Is this theorem concerning the change of bases and my proof thereof correct?

I seek verification of my proof for this change of bases theorem. Any help would be greatly appreciated! The theorem is stated below: Let $V, W$ be finite-dimensional vector spaces over a field $...
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Elementary Divisors of a generated group

Let $H \subset \mathbb Z^4$ be the group generated by the elements (3, 9, 3, 0) and (4, 2, 0, 2). Find the rank and the elementary divisors of $A := \mathbb Z^4 / H$. I know how to find this when ...
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When a family of subspaces accept a suitable set of basis using expansion basis?

Recently, I want to prove a structure theorem of submodule of finitely generated module over PID. By several process, it reduces to the following problem in linear algebra. Given a linear space $V$...
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1answer
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Does this change of basis exist?

Let $A$ be a unital associative $\mathbb{C}$ algebra and let $M$ be a 2-dimensional $A$-module with basis $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$. $A$ has two generators $\left\{a_{1}, a_{2}\...
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1answer
29 views

Babilonic notation to decimal notation. Example $1;12 \cdot 15$

I'm currently working in a program that convert numbers in babilonic notation into decimal numbers. The problem I have is that the example and requirements described by the teacher deliver numbers in ...
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2answers
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Orthonormal basis containing a given vector

What is the easiest way to find an orthonormal basis that contains a given vector in $\mathbb{R}^n$ ? I am looking for the change-of-basis matrix from the standard Euclidean basis containing the axis-...
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1answer
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If P is a transition matrix from basis A to B, Q is a transition matrix from B to C, then PQ is a transition matrix from A to C.

Problem: Let $B = (w_1,...,w_n)$ and $C = (u_1,...,u_n)$ be bases in linear space V. $P$ is a transition matrix from basis $A$ to $B$. $Q$ is a transition matrix from basis $B$ to $C$. Prove that $PQ$ ...
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5answers
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Why is 1, x and $x^2$ linearly independent?

We say that (1,x,$x^2$) span the set of polynomials of degree 2? But why do we say they are linearly independent? How do you define linear independence of functions like $f(x) = x^2$ and g(x) = x? ...
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Change of a basis for polynomials

We have given two bases ${p_0,..,p_n}$ and ${q_0,..,q_n}$ of a polynomial space. The interpolation points are given with $x_i, i = 0,..,n$ and two matrices $G$ and $H$ are defined with $G_{ij = p_j(...
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Gaussian elimination in vector spaces

I've been working on a set of problems while learning matrix operations as well as vector spaces and subspaces. But now I have some doubts that go outside the general rule of thumb and I'm unable to ...
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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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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
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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
25 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
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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
43 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
24 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
65 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
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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 ...