Are the following vectors in $\displaystyle P_2$ linearly dependent? If yes, express one vector as a linear combination of the rest.

$\displaystyle {3t+1,3t^2+1,2t^2+t+1}$

This forms the matrix $\displaystyle

\[

\left[ {\begin{array}{ccc}

0 & 3 & 2 \\

3 & 0 & 1 \\

1 & 1 & 1 \\

\end{array} } \right]

\]$ which is row equivalent to $\displaystyle \[

\left[ {\begin{array}{ccc}

1 & 0 & \frac{1}{3} \\

0 & 1 & \frac{2}{3} \\

0 & 0 & 0 \\

\end{array} } \right]

\]$

So $\displaystyle c_1=\frac{r}{3},c_2=\frac{2r}{3}, c_3=-r$ is a solution for the homogeneous system, where r is any real number.

Let r = 1, then $\displaystyle \frac{1}{3}P_1 + \frac{2}{3}P_2-1=0$

Solving for $\displaystyle P_1=\frac{1}{3}-\frac{2}{3}P_2$

But when I check it doesn't work: $\displaystyle 3t+1 \not= \frac{1}{3}-\frac{2}{3}(3t^2+1)$