How to Find a New Basis for a Polynomial Representation in \(\mathcal{P}_3\)?

[e(ho0n3]

Homework Statement
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.

$$\text{Rep}_B(1 - x + 3x^2 - x^3) = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \end{pmatrix}_B$$

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

$$\text{Rep}_D(1 - x + 3x^2 - x^3) = \begin{pmatrix} 1 \\ 0 \\ 2 \\ 0 \end{pmatrix}_D$$

The attempt at a solution
I've noticed that

$$1 - x + 3x^2 - x^3 = 1 - x + x^2 + x^3 + 2(x^2 - x^3)$$

so the first and third component of $D$ could be $1 - x + x^2 + x^3$ and $x^2 - x^3$ respectively. I can guess a possible second and fourth component and then check $D$ to determine if it is a basis. Is there an easier way of accomplishing this?

 

[mighty2000]

Thank you for your question. It is great that you have already made some observations about the given polynomial and its representation in \mathcal{P}_3 with basis B. You are correct in noticing that the first and third components of D could be 1 - x + x^2 + x^3 and x^2 - x^3, respectively. However, instead of guessing the second and fourth components, we can use a more systematic approach to find a basis D that gives the desired representation.

First, let's rewrite the polynomial as a linear combination of the elements in B:

1 - x + 3x^2 - x^3 = 1(1 + x) + (-1)(1 - x) + 3(x^2 + x^3) + (-1)(x^2 - x^3)

= 1(1 + x) + (-1)(1 - x) + 3(x^2 + x^3) + (-1)(x^2 - x^3)

= (1 - 1 + 3x^2 - x^2) + (x - x - 3x^3 + x^3)

= 3x^2 - 2x^3

Next, we can express this linear combination as a matrix multiplication:

\begin{pmatrix} 1 & -1 & 3 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 + x \\ 1 - x \\ x^2 + x^3 \\ x^2 - x^3 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \end{pmatrix}_B

Notice that the matrix on the left is the change of basis matrix from B to the standard basis of \mathcal{P}_3. Therefore, in order to find a basis D that gives the desired representation, we need to find the inverse of this matrix and apply it to the vector on the right.

To find the inverse of the matrix, we can use elementary row operations to reduce it to the identity matrix. The resulting matrix will be the inverse of the original matrix. I will leave it to you to perform