In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

The vectorsof the given polynomials can be written as follows:

\(1 + 2{t^3} \equiv \left( {\begin{array}{*{20}{c}}1\\0\\0\\2\end{array}} \right)\),\(\left( {2 + t - 3{t^2}} \right) \equiv \left( {\begin{array}{*{20}{c}}2\\1\\{ - 3}\\0\end{array}} \right)\),\(\left( { - t + 2{t^2} - {t^3}} \right) \equiv \left( {\begin{array}{*{20}{c}}0\\{ - 1}\\2\\{ - 1}\end{array}} \right)\)

The matrix formed by using the vectors of the polynomials is:

\(A = \left( {\begin{array}{*{20}{c}}1&2&0\\0&1&{ - 1}\\0&{ - 3}&2\\2&0&{ - 1}\end{array}} \right)\)

\(\left( {\begin{array}{*{20}{c}}1&2&0\\0&1&{ - 1}\\0&{ - 3}&2\\2&0&{ - 1}\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\\0&0&0\end{array}} \right)\)

As the matrix has a pivot in each column, its columns are linearly independent.

So, the polynomials are linearly independent.