# The coordinate vector of \displaystyle{\left[{\mathbf{{{p}}}}

The coordinate vector of relative to the basis

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Nathaniel Kramer

Any arbitrary vector in ${\mathbb{P}}_{\mathbb{2}}$ can be written as,

Thus, for the vector can be written as,
1)
On comparing the terms of both the sides in the above equation (1) gives the following equations.

The representation of the linear system in a matrix is,
$\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}a\\ c\\ d\end{array}\right]=\left[\begin{array}{c}6\\ 3\\ -1\end{array}\right]$
The solution of the above system gives as by solving the augmented matrix
$\left[\begin{array}{cccc}1& 1& 0& 6\\ 1& 0& 1& 3\\ 0& 1& 1& -1\end{array}\right]$
Use Elementary row transformations to reduce the augmented matrix to row reduced Echelon form.
Step 1:

Step 2

Step 3:

Step 4: