Answer
There are two zero rows on the augmented reduced matrix, hence the given set of vectors is linearly dependent
Work Step by Step
Given Polynomial:
$( 2 - t ) ^ { 3 } , ( 3 - t ) ^ { 2 } , 1 + 6 t - 5 t ^ { 2 } + t ^ { 3 }$
we need to convert the polynomials into vectors;
By expanding we get;
$\begin{matrix} ( 2 - t ) ^ { 3 } = 8 - 12 t + 6 t ^ { 2 } - t ^ { 3 } \\ ( 3 - t ) ^ { 2 } = 9 - 6 t + t ^ { 2 } \\ 1 + 6 t - 5 t ^ { 2 } + t ^ { 3 } = 1 + 6 t - 5 t ^ { 2 } + t ^ { 3 } \end{matrix}$
forming the vectors
$8 - 12 t + 6 t ^ { 2 } - t ^ { 3 }=\begin{bmatrix}8\\-12\\6\\-1\end{bmatrix}$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,9 - 6 t + t ^ { 2 } =\begin{bmatrix}9\\-6\\1\\0\end{bmatrix}$
$\,\,\,\,\,\,1 + 6 t - 5 t ^ { 2 } + t ^ { 3 }=\begin{bmatrix}1\\6\\-5\\1\end{bmatrix}$
We combine an Augmented Matrix form the vectors and row reduce;
$\begin{bmatrix}8&9&1&0\\{-12}&{-6}&6&0\\6&1&-5&0\\-1&0&1&0\end{bmatrix}\sim\begin{bmatrix}1&0&-1&0\\0&1&1&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$
This show that there are two zero rows, hence the given set of vectors is linearly dependent
