Answer
There are no free variables, hence the given set is linearly independent
Work Step by Step
Given Polynomial:
$1 - 2 t ^ { 2 } - t ^ { 3 } , t + 2 t ^ { 3 } , 1 + t - 2 t ^ { 2 }$
we need to convert the polynomials into vectors;
$1 - 2 t ^ { 2 } - t ^ { 3 }=\begin{bmatrix}1\\0\\-2\\1\end{bmatrix}$
$0 + t + 2 t ^ { 3 } =\begin{bmatrix}0\\1\\0\\2\end{bmatrix}$
$1 + t - 2 t ^ { 2 }=\begin{bmatrix}1\\1\\-2\\0\end{bmatrix}$
We combine an Augumented Matrix form the vectors
$\begin{bmatrix}1&0&1&0\\{ 0}&{1}&1&0\\-2&0&-2&0\\-1&2&0&0\end{bmatrix}\sim\begin{bmatrix}1&0&1&0\\0&1&1&0\\0&0&-1&0\\0&0&0&0\end{bmatrix}$
This show that there are no free variables, hence the given set is linearly independent
