 # 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 generals336 2021-09-13 Answered

Determine if the following set of polynomials fors a basis of ${P}_{3}$
${p}_{1}=3+7t,$

${p}_{2}=5+t-2{t}^{3},$

${p}_{3}=t-2{t}^{2},$

${p}_{4}=1+16t-6{t}^{2}+2{t}^{3}$

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We have given the polynomials
${p}_{1}=3+7t,$

${p}_{2}=5+t-2{t}^{3},$

${p}_{3}=t-2{t}^{2},$

${p}_{4}=1+16t-6{t}^{2}+2{t}^{3}$
The coordinate mapping gives the following coordinate vectors

Write the above vectors as a column matrix and reduce the matrix into reduced row echelon form as follows.
$\left[\begin{array}{cccc}3& 5& 0& 1\\ 7& 1& 1& 16\\ 0& 0& -2& -6\\ 0& -2& 0& 2\end{array}\right]\to \left[\begin{array}{cccc}3& 5& 0& 1\\ 1& -9& 1& 14\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}3& 5& 0& 1\\ 1& -9& 1& 14\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& -9& 1& 14\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& -9& 1& 14\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& 0& 0& 2\\ 3& 5& 0& 1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 5& 0& -5\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$
$\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 5& 0& -5\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& -1\\ 0& 0& 1& 3\\ 0& 1& 0& -1\end{array}\right]$