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**1. The problem statement**

find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector [tex]x = \begin{pmatrix}-1\\-13\\

9\\

\end{pmatrix}

\in\mathbb R[/tex]

if [tex]{β= \begin{pmatrix}-1\\4\\

-2\\

\end{pmatrix},\begin{pmatrix}3\\-1\\

-2\\

\end{pmatrix},\begin{pmatrix}2\\-5\\

1\\

\end{pmatrix}}[/tex] and [tex]{γ= \begin{pmatrix}3\\-1\\

-2\\

\end{pmatrix},\begin{pmatrix}1\\4\\

-2\\

\end{pmatrix},\begin{pmatrix}2\\-5\\

1\\

\end{pmatrix}}[/tex]

find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector [tex]x = \begin{pmatrix}-1\\-13\\

9\\

\end{pmatrix}

\in\mathbb R[/tex]

if [tex]{β= \begin{pmatrix}-1\\4\\

-2\\

\end{pmatrix},\begin{pmatrix}3\\-1\\

-2\\

\end{pmatrix},\begin{pmatrix}2\\-5\\

1\\

\end{pmatrix}}[/tex] and [tex]{γ= \begin{pmatrix}3\\-1\\

-2\\

\end{pmatrix},\begin{pmatrix}1\\4\\

-2\\

\end{pmatrix},\begin{pmatrix}2\\-5\\

1\\

\end{pmatrix}}[/tex]

**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?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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