# 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 $\mathbb{R}^2$...
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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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### 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 ...
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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 ...
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### 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 ...
### 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 ...