Questions

The problem you presented involves dealing with vectors, transformations, and bases in the context of the vector space of polynomials, Pₙ. Here’s a systematic approach to solve each part of the problem given:

1. Finding the 2x5 matrix $A: \mathbb{R}^P \to P_n$:

Since you're given that the standard basis for S is {1, 4, 22, 23, 4}, let's assume there are some typos or misleading notation because it's difficult to interpret. So, let’s rewrite what seems to be necessary interpretations:

• Typically, the standard basis for Pₙ would be something like {1, x, x², ..., x^n}, for polynomials up to degree n. Let’s assume the correct basis might be {1, x, x², x³, x⁴} for n = 4.

2. Finding the 2x5 matrix $A$:

You need to find the matrix A describing the transformation of polynomials represented by $P_S$. Assuming $P_S$ transforms a vector $[a_0, a_1, a_2, a_3, a_4]$ (coefficients with respect to the standard basis) into a polynomial, the transformation would map:

$P_S : \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix} \to a_0 * 1 + a_1 * x + a_2 * x^2 + a_3 * x^3 + a_4 * x^4$

However, your original problem isn't clear about what transformation $P_S$ exactly is. Let's assume you meant finding coordinates concerning another basis.

Let’s consider the matrix representing the transformation that maps coefficients from the basis B to the standard basis of P₄. Suppose the given set in the matrix relates those conversions.

Assume the supposedly new basis of P₄ is { S₁, S₂, S₃, S₄, S₅ ], where these represents polynomials associated (1,4,22,23,4):

$S_1 = 1, S_2 = x, S_3 = x^2, S_4 = x^3, S_5 = x^4$

The transformation matrix now would be straightforwardly the identity over 5 terms (Due to linear bases equivalency):