Let's break down problem step by step: --- ## Restatement Given distinct points \( x_, x_1, \ldots, x_n \) and corresponding values \( y_, y_1, \ldots, y_n \), we want to find the unique polynomial \( p_n(x) \) of degree \( \leq n \) such that \( p_n(x_i) = y_i \) for all \( i = , 1, \ldots, n \). We are told this can be written as the matrix equation \( W A = Y \), where \( W \) is the Vandermonde matrix. --- ## a) **Find the Matrix \( W \)** The polynomial of degree \( \leq n \) can be written as: \[ p_n(x) = a_ + a_1 x + a_2 x^2 + \ldots + a_n x^n \] where \( a_, a_1, \ldots, a_n \) are the coefficients. For each \( x_i \), we have: \[ p_n(x_i) = a_ + a_1 x_i + a_2 x_i^2 + \ldots + a_n x_i^n = y_i \] Writing this for all \( i = , 1, \ldots, n \) yields the system: \[ \begin{bmatrix} 1 & x_ & x_^2 & \cdots & x_^n \\ 1 & x_1 & x_1^2 & \cdots & x_1^n \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^n \\ \end{bmatrix} \begin{bmatrix} a_ \\ a_1 \\ \vdots \\ a_n \end{bmatrix} = \begin{bmatrix} y_ \\ y_1 \\ \vdots \\ y_n \end{bmatrix} \] The matrix on the left is the **Vandermonde matrix \( W \)**. ### **Explicitly:** \[ \boxed{ W = \begin{bmatrix} 1 & x_ & x_^2 & \cdots & x_^n \\ 1 & x_1 & x_1^2 & \cdots & x_1^n \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^n \\ \end{bmatrix} } \] --- ## b) **Why is the Vandermonde matrix \( W \) nonsingular?** **Key Fact:** A square Vandermonde matrix \( W \) is nonsingular (invertible) **if and only if all the \( x_i \) are distinct**. **Reason:** - The determinant of a Vandermonde matrix is: \[ \det(W) = \prod_{ \leq i < j \leq n} (x_j - x_i) \] - If all \( x_i \) are distinct, every term \( (x_j - x_i) \neq \), so the product is nonzero. - Thus, **\( W \) is invertible** (nonsingular). **No algebra needed**—just knowing the determinant formula and the distinctness of the \( x_i \)'s! --- ### **Summary** - **a)** The Vandermonde matrix \( W \) is as described above. - **b)** \( W \) is nonsingular because the \( x_i \) are distinct, and thus the determinant is nonzero. Let me know if you want to see a worked example!