Linearly independent polynomials; answer not checking out.

**Jskid** · February 28th 2011, 08:38 PM

Are the following vectors in $P_2$ linearly dependent? If yes, express one vector as a linear combination of the rest.
${3t+1,3t^2+1,2t^2+t+1}$

This forms the matrix $ \[ \left[ {\begin{array}{ccc} 0 & 3 & 2 \\ 3 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array} } \right] \]$ which is row equivalent to $ \left[ {\begin{array}{ccc} 1 & 0 & \frac{1}{3} \\ 0 & 1 & \frac{2}{3} \\ 0 & 0 & 0 \\ \end{array} } \right] $
So $c_1=\frac{r}{3},c_2=\frac{2r}{3}, c_3=-r$ is a solution for the homogeneous system, where r is any real number.
Let r = 1, then $\frac{1}{3}P_1 + \frac{2}{3}P_2-1=0$
Solving for $P_1=\frac{1}{3}-\frac{2}{3}P_2$
But when I check it doesn't work: $3t+1 \not= \frac{1}{3}-\frac{2}{3}(3t^2+1)$