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We are given the following problem involving a set of polynomials:

### Problem Setup:
Let \( S = \{ x^3 - x^2 - x + 1, -x + 1, 2x^3 - x^2 - x, 15x^3 - 8x^2 - 13x + 12 \} \).

We need to address the following parts:

### (a) **Determine a maximal linearly independent subset \( A \) of \( S \).**

To determine the maximal linearly independent subset, we treat these polynomials as vectors in a vector space of polynomials of degree at most 3. We will express the polynomials as vectors in terms of their coefficients:

1. \( p_1(x) = x^3 - x^2 - x + 1 \) corresponds to the vector \( \begin{pmatrix} 1 \\ -1 \\ -1 \\ 1 \end{pmatrix} \).
2. \( p_2(x) = -x + 1 \) corresponds to the vector \( \begin{pmatrix} 0 \\ 0 \\ -1 \\ 1 \end{pmatrix} \).
3. \( p_3(x) = 2x^3 - x^2 - x \) corresponds to the vector \( \begin{pmatrix} 2 \\ -1 \\ -1 \\ 0 \end{pmatrix} \).
4. \( p_4(x) = 15x^3 - 8x^2 - 13x + 12 \) corresponds to the vector \( \begin{pmatrix} 15 \\ -8 \\ -13 \\ 12 \end{pmatrix} \).

Now, we will check for linear dependence by forming a matrix with these vectors as columns and reducing it to row echelon form.

### (b) **Hence calculate \( \dim(\text{span}(S)) \) and justify whether \( \text{span}(S) = \mathcal{P}_3 \).**

The dimension of the span of \( S \) is simply the number of linearly independent vectors in \( S \). We will determine this by the rank of the matrix from part (a). To check whether \( \text{span}(S) = \mathcal{P}_3 \), we need to verify if the dimension of the span equals 4, which is the dimension of the vector space of polynomials of degree at most 3, \( \mathcal{P}_3 \).

### (c) **Is \( A \) a minimal spanning set for \( \text{span}(S) \)? Justify your answer.**

A minimal spanning set has no redundant vectors. If the set \( A \) found in part (a) is linearly independent and spans the same space as \( S \), then it is a minimal spanning set.

---

I will now perform the matrix analysis to solve part (a).### (a) **Maximal Linearly Independent Subset \( A \) of \( S \):**

The matrix formed by the coefficients of the polynomials in \( S \) has full rank \( 4 \). This means all four polynomials are linearly independent. Therefore, the maximal linearly independent subset \( A \) is the entire set \( S \):
\[ A = \{ x^3 - x^2 - x + 1, -x + 1, 2x^3 - x^2 - x, 15x^3 - 8x^2 - 13x + 12 \}. \]

### (b) **Dimension of \( \text{span}(S) \):**

Since all four vectors are linearly independent, the dimension of \( \text{span}(S) \) is 4:
\[
\dim(\text{span}(S)) = 4.
\]
This shows that \( \text{span}(S) = \mathcal{P}_3 \), because \( \mathcal{P}_3 \) is the space of all polynomials of degree at most 3, which also has dimension 4.

### (c) **Is \( A \) a minimal spanning set for \( \text{span}(S) \)?**

Yes, \( A \) is a minimal spanning set for \( \text{span}(S) \), because it is both linearly independent and spans the entire space. There are no redundant vectors in \( A \).

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Would you like further details or have any questions? Here are some related questions:

1. What does it mean for a set of vectors to be linearly independent?
2. How is the dimension of a vector space defined?
3. Can the span of fewer than 4 polynomials ever equal \( \mathcal{P}_3 \)?
4. What is the significance of row-reduction in determining linear independence?
5. How would the analysis change if one of the polynomials in \( S \) were dependent?

**Tip:** When finding the dimension of a span, always compare it with the dimension of the ambient space to understand the relationship.

Let S = {x^3 - x^2 - x + 1, -x + 1, 2x^3 - x^2 - x, 15x^3 - 8x^2 - 13x + 12}, then determine: (a) A maximal linearly independent subset A of S. (b) The dimension of span(S) and whether span(S) equals P_3. (c) Whether A is a minimal spanning set for span(S).

Learn how to determine the linear independence of a set of polynomials and calculate the dimension of the span. This problem includes checking whether a given set spans the vector space P_3 and finding a minimal spanning set.

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