Answer
See answer below
Work Step by Step
We are given:
$P_1(x)=2+x^2$
$P_2(x)=4-2x+3x^2$
$P_3(x)=1+x$
The augmented matrix of this system is:
$\begin{bmatrix}
2 & 4 &1 | 0\\
0& -2 & 1 | 0\\
1 & 3 & 0 | 0
\end{bmatrix} \approx \begin{bmatrix}
2 & 4 &1 | 0\\
0& -2 & 1 | 0\\
0 & -2 & 1 | 0
\end{bmatrix} \approx \begin{bmatrix}
2 & 4 &1 | 0\\
0& -2 & 1 | 0\\
0 & 0 & 0 | 0
\end{bmatrix}$
$\rightarrow rank(A)=2$
$ dim (P_2)=3$
$\rightarrow rank (A) \lt dim (P_2)$
Hence, $P_1. P_2, P_3$ don't span P_2(R). Each of two polynominals from the independent set of vectors spans the same subspace of V.
