If B is the standard basis of the space $P_3$ of polynomials, then let $B = \{1, t, t^2, t^3\}$. Use coordinate vectors to test the linear independence of the set of polynomials below. Explain your work.
$1 - 9t^2 - t^3$, $t + 4t^3$, $1 + t - 9t^2$
Write the coordinate vector for the polynomial $1 - 9t^2 - t^3$.
$(1, 0, -9, -1)$
Write the coordinate vector for the polynomial $t + 4t^3$.
$(0, 1, 0, 4)$
Write the coordinate vector for the polynomial $1 + t - 9t^2$.
$(1, 1, -9, 0)$
To test the linear independence of the set of polynomials, row reduce the matrix which is formed by making each coordinate vector a column of the matrix. If possible, write the matrix in reduced echelon form.
$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ -9 & 0 & -9 \\ -1 & 4 & 0 \end{bmatrix}$
