## Expressing a Vector in Terms of Given Vectors in R3

We can write a vector as a linear combination of n linearly independent vectors in an n dimensional space.
Suppose we wish to write the vector
$\begin{pmatrix}1\\4\\-3\end{pmatrix}$
in terms of the vectors
$\begin{pmatrix}1\\-2\\5\end{pmatrix} , \: \begin{pmatrix}2\\-3\\0\end{pmatrix} , \: \begin{pmatrix}0\\1\\3\end{pmatrix}$

Then
$\begin{pmatrix}1\\4\\-3\end{pmatrix} = a \begin{pmatrix}1\\-2\\5\end{pmatrix} +b \begin{pmatrix}2\\-3\\0\end{pmatrix} +c \begin{pmatrix}0\\1\\3\end{pmatrix}$

This is equivalent to the system of equations
$1=a+2b$

$4=-2a-3b+c$

$-3=5a+c$

This system has solutions
$a=- \frac{17}{11} ,\: b=\frac{14}{11} , \: c=\frac{52}{11}$

Then
$\begin{pmatrix}1\\4\\-3\end{pmatrix} = - \frac{17}{11} \begin{pmatrix}1\\-2\\5\end{pmatrix} + \frac{14}{11} \begin{pmatrix}2\\-3\\0\end{pmatrix} + \frac{52}{11} \begin{pmatrix}0\\1\\3\end{pmatrix}$

We can write define a basis
$\{ \mathbf{e_1}, \: \mathbf{e_2}, \: \mathbf{e_3} \}$
then
$\mathbf{e_1} = \begin{pmatrix}1\\-2\\5\end{pmatrix} , \: \mathbf{e_2}= \begin{pmatrix}2\\-3\\0\end{pmatrix} , \: \mathbf{e_3} =\begin{pmatrix}0\\1\\3\end{pmatrix}$

Then
$\begin{pmatrix}1\\4\\-3\end{pmatrix} = - \frac{17}{11} \mathbf{e_1} + \frac{14}{11} \mathbf{e_2} + \frac{52}{11} \mathbf{e_3}$