Determine if the following set of polynomials fors a basis of P_3 , p_1=3+7t, p_2=5+t-2t^3 ,p_3=t-2t^2, p_4=1+16t-6t^2+2t^3

We have given the polynomials
${\displaystyle p_1=3+7t,}$

$p_2=5+t-2t^3,$

$p_3=t-2t^2,$

$p_4=1+16t-6t^2+2t^3$
The coordinate mapping gives the following coordinate vectors
$\left[\begin{array}{c}3\\ 7\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}5\\ 1\\ 0\\ -2\end{array}\right],\left[\begin{array}{c}0\\ 1\\ -2\\ 0\end{array}\right]and\left[\begin{array}{c}1\\ 16\\ -6\\ 2\end{array}\right]$
Write the above vectors as a column matrix and reduce the matrix into reduced row echelon form as follows.
$\left[\begin{array}{cccc}3& 5& 0& 1\\ 7& 1& 1& 16\\ 0& 0& -2& -6\\ 0& -2& 0& 2\end{array}\right]\to \left[\begin{array}{cccc}3& 5& 0& 1\\ 1& -9& 1& 14\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}3& 5& 0& 1\\ 1& -9& 1& 14\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& -9& 1& 14\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& -9& 1& 14\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& 0& 0& 2\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 5& 0& -5\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 5& 0& -5\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& -1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$