# Math Help - Linearly independent polynomials; answer not checking out.

1. ## Linearly independent polynomials; answer not checking out.

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: $3t+1 \not= \frac{1}{3}-\frac{2}{3}(3t^2+1)$

2. $P_1=2t^2+t+1,P_2=3t+1,P_3=3t^2+1$.
$P_1=\frac{1}{3}P_2+\frac{2}{3}P_3$.

3. $2t^2+t+1=a(3t+1)+b(3t^2+1)\Rightarrow \left\{\begin{matrix}
3b=2 \\
3a=1 \\
a+b=1
\end{matrix}\right.$
$\Rightarrow a=\frac{1}{3},b=\frac{2}{3}$