# Thread: Find the coordinate vector

1. ## Find the coordinate vector

Let $\beta$ = { $v_1, v_2, v_3$} be the basis for $R^3$ where the v's are
$[1, 2, -2]^T$, $[-1, -1, 3]^T$ $[2, 6, -1]^T$ respectively.

Find the coordinate vector with respect to $\beta$ for the vector v = $[3, 5, -6]^T$

2. You have two options: build the matrix that changes the bases (i don't know his english name ), or do a linnear equation system. do you need more help?

3. Originally Posted by wopashui
Let $\beta$ = { $v_1, v_2, v_3$} be the basis for $R^3$ where the v's are
$[1, 2, -2]^T$, $[-1, -1, 3]^T$ $[2, 6, -1]^T$ respectively.

Find the coordinate vector with respect to $\beta$ for the vector v = $[3, 5, -6]^T$

It must be $\begin{pmatrix}\;3\\\;5\\\!\!-6\end{pmatrix}=a\begin{pmatrix}\;1\\\;2\\\!\!-2\end{pmatrix}$ $+b\begin{pmatrix}\!\!-1\\\!\!-1\\\;3\end{pmatrix}+c\begin{pmatrix}2\\6\\\!\!\!-1\end{pmatrix}$ , with $a,b,c,\in\mathbb{R}\Longleftrightarrow\begin{array }{l}\;\;a-b+2c=\;\;3\\2a-b+6c=\;\;5\\\!\!\!\!-2a+3b-c=\!-6\end{array}$ . Well, solve this system now.

Tonio