# Linear algebra ordered basis problem

1. The problem statement

find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector $$x = \begin{pmatrix}-1\\-13\\ 9\\ \end{pmatrix} \in\mathbb R$$

if $${β= \begin{pmatrix}-1\\4\\ -2\\ \end{pmatrix},\begin{pmatrix}3\\-1\\ -2\\ \end{pmatrix},\begin{pmatrix}2\\-5\\ 1\\ \end{pmatrix}}$$ and $${γ= \begin{pmatrix}3\\-1\\ -2\\ \end{pmatrix},\begin{pmatrix}1\\4\\ -2\\ \end{pmatrix},\begin{pmatrix}2\\-5\\ 1\\ \end{pmatrix}}$$

3. The Attempt at a Solution

i read my notes and as i understood it, an ordered basis is the linear combination that you use to obtain a specific vector in a vector space. im not clear on the beta and gamma coordinates,
and i cant understand why the β and γ basis includes 3 vectors? im thinking on the lines that x is obtained through a combination between the β coordinates and the given β , but that does not get me anywhere. please someone point me in the right direction! thank you

edit : ok i understand that β times the β coordinates would give the vector before the transformation, and γ
times the γ coordinates give the vector after transformation. but what exactly is x then? the vector before transformation or the vector after transformation?

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ok i have finally managed to get an answer for this question but im not sure of it at all, this is what i did

[β]x(a,b,c,d) = x where (a,b,c,d) is the beta coordinate of x

and i solved this equation and ended up getting some values for a b c and d which i wrote down as the β coordinates. is this wrong?

our instructor solved this problem so anyone wanna know the solution just let me know :)

our instructor solved this problem so anyone wanna know the solution just let me know :)
I would like to know.