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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.

Linear transformation between finite dimensional vector spaces, $T:V\rightarrow W$, has a matrix representation $A_T$ with respect to given basis $B_V=\{v_1,\cdots,v_n\},\ B_W=\{w_1,\cdots,w_m\}$ Here we have another $A_T'$ with respect to basis $B_V'=\{v_1',\cdots,v_n'\},\ B_W'=\{w_1',\cdots,w_m'\}$ Also we have coordinate isomorphisms $P_V,\ P_W$ : $$ P_V v_i=v_i',\ P_W w_i=w_i' $$ So we have $P_WA_T'P_V^{-1}=A_T$

1175 questions
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Change of basis matrix with different dimensions

Let's say I have 2 bases $A$ from $R^x$ and $B$ from $R^y$. Is it possible to compute the change of basis matrix from $A$ to $B$ if $x \ne y$ ? If $x = y$, it's easy, but if it is possible for $x \ne y$, how can I do it?
Daniel
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Topological basis on $\mathbb R^2$

Let $\mathcal B$ be the collection of all partial opened rectangles $[a,b)\times [c,d)$, where $a \lt b, c \lt d$. Show that $\mathcal B$ is a topological basis on $\mathbb R^2$. Is the topology generated by $\mathcal B$ finer than the usual…
Verse
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Calculating a vector in a basis

Consider the space of polynomials of degree equal or less than 2. The canonical basis of this space is $<1,x,x^2>$ and another basis can be $<1, 1+x, 1-x+x^2>$. Calculating the change of basis matrix $A_{b1,bcan}$ and apply it to the vector…
Granger Obliviate
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Change of basis

The matrix $A$ has the eigenvalues $\lambda_1 = 1$ and $\lambda_2 = 4$ as well as the eigenvectors $v_1 = (2, 3)$ to $\lambda_1$ and $v_2 = (1, 2)$ to $\lambda_2$. From this information determine the matrix $A$ with respect to the canonical basis $(…
A O
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Change of basis when the basis aren't square

If I have a basis $$B=\{(1,2,0,1), (0,8,0,7), (0,0,1,0)\}$$ and an orthogonal basis (which I obtained from $B$) $$W = \{(1,2,0,1), (-23, 2, 0, 19), (0,0,1,0)\}$$ How would I calculate the change of basis matrix from $B$ to $W$? (I'd usually use the…
daniel
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find a basis of $\ker T$ and find a basis of $\mathrm{Im} T$. You may assume that $T$ is linear.

let $T:\Bbb R^3 → \Bbb R^3$; $T (x, y, z)= (x+y, x+y, 0)$ for $(x, y, z) \in\Bbb R^3$
user834488
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Change of basis related to conics

I was trying to solve some change of basis exercises, so I've done them and I want you to tell me if I've done it correctly. 1) Given the coordinates system $(O'; X'' Y'')$ asociated to the basis $B=[{b_1=\frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}};…
AaronTBM
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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 the following format $$1;12 \cdot 15$$ That would…
mraz
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Writing the equation of a plane with a different basis.

Given the plane $-x-y+2z=1$ in the basis $\{(1,0,0),(0,1,0),(0,0,1)\}$ I want to calculate the equation of this plane respect the basis $\{\frac{1}{\sqrt2}(-1,1,0),\frac{1}{\sqrt3}(-1,-1,-1),\frac{1}{\sqrt6}(-1,-1,2)\}$. I know that if I want to…
John Keeper
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Is this an ordered basis

I was just having trouble with this question. If $S=[e_1-e_2,e_3]$ where $e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1)$ Would the list $S$ be an ordered basis of $\Bbb R^3$ Help would really be appreciated on this one guys
Elena
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Linear transformation with change of basis

If I have a linear transformation $T$ and the basis $b$ and $a$, when I have $[T]_{b}^{a}$ It means that I'm doing the transformation starting on the basis $a$ and the result is given with respect of basis $b$?! If so, it is correct to apply first…
Bruno Reis
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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.
Ifex
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