Exercise 15

• In $\mathcal{P}_{3}$ with basis $B=\langle 1+x,1-x,x^{2}+x^{3},x^{2}-x^{3}\rangle$ we have this representation.

  $${\rm Rep}_{B}(1-x+3x^{2}-x^{3})=\begin{pmatrix}0\\ 1\\ 1\\ 2\end{pmatrix}_{B}$$

  Find a basis $D$ giving this different representation for the same polynomial.

  $${\rm Rep}_{D}(1-x+3x^{2}-x^{3})=\begin{pmatrix}1\\ 0\\ 2\\ 0\end{pmatrix}_{D}$$

• State and prove that we can change any nonzero vector representation to any other.

Hint. The proof of Lemma 1.5 is constructive — it not only says the bases change, it shows how they change.