Answer
The set of polynomials are linearly dependent.
Work Step by Step
Step 1:
Write the coordinate vectors produced by the coordinate mapping of the set (solve eventual calculations)
${(1-t^2), t-2t^2+t^3, (1-t)^3}$
$\Leftrightarrow$
${1-2t+t^2, t-2t^2+t^3, 1-3t+3t^2-t^3}$
that is
(1 -2 1 0 ), (0 1 -2 1), (1 -3 3 -1)
Step 2:
Write the coordinate vectors as a column matrix and test for linear dependency (A$\mathbf{x}=\mathbf{0}$)
\begin{bmatrix}
1 & 0 & 1 & 0 \\
-2 & 1 & -3 & 0 \\
1 & -2 & 3 & 0 \\
0 & 1 & 1 & 0
\end{bmatrix}
$\Leftrightarrow$
\begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
$\implies$ The columns of the matrix are linearly dependent since there is a free variable (there is not a pivot position in every column).
Hence the corresponding polynomials are also linearly dependent.
