Determine coordinates for $p=7-12t-8t^2+12t^3$ in another basis.

Question

Let $S=\{1,2t,-2+4t^2,-12t+8t^3\}$ be a set of polynomials in $P_3$. Show that these polynomials make up a basis for $P_3$ and determine the coordinates for $p=7-12t-8t^2+12t^3$ in this basis.

The first part of the problem was easy, showing that they make up a basis by showing that the polynomials are linearly independant and since $\dim(P_3)=\dim(S)=4,$ they can span $P_3$.

But how do I determine the coordinates for $p=7-12t-8t^2+12t^3$ in this basis?

Answer 1

HINT

As a general method

• write the matrix $M$ from the basis S to the standard basis; M has for columns the component of the basis vectors with respect to the standard basis, that is

$$M=\begin{bmatrix}1&0&-2&0\\0&2&0&-12\\0&0&4&0\\0&0&0&8 \end{bmatrix}$$

• then if $v_S$ is a vector in the $S$ basis we have $v=Mv_S$ where $v$ is expressed in the standard basis
• then $v_s=M^{-1}v$

Answer 2

Let $p_0=1$, $p_1=2t$, $p_2=4t^2-2$, and $p_3=8t^3-12t$ be the basis vectors. For $p=12t^3-8t^2-12t+7$, it's convenient to note first that

$$2p-3p_3=(24t^3-16t^2-24t+14)-(24t^3-36t)=-16t^2+12t+14$$

From there you can see that

$$2p-3p_3+4p_2=(-16t^2+12t+14)+(16t^2-8)=12t+6$$

Can you see how to continue?