Question

Let \(R\) be an integral domain, \(u_{0}, \ldots, u_{n-1} \in R\), and \(V=\operatorname{VDM}\left(u_{0}, \ldots, u_{n-1}\right) \in R^{n \times n}\). Prove that $$ \operatorname{det} V=\prod_{1 \leq j

Short answer

The determinant of the Vandermonde matrix is the product of all differences \((u_i - u_j)\) for \(1 \leq j < i \leq n\).

Step-by-step solution

Define the Vandermonde Matrix

The given problem involves the determinant of a Vandermonde matrix. A Vandermonde matrix, \( V = \operatorname{VDM}(u_0, \ldots, u_{n-1}) \), is an \( n \times n \) matrix where the \( i \)-th row is \( [1, u_i, u_i^2, \ldots, u_i^{n-1}] \). The general form of the Vandermonde matrix is:\[V = \begin{pmatrix}1 & u_0 & u_0^2 & \cdots & u_0^{n-1} \1 & u_1 & u_1^2 & \cdots & u_1^{n-1} \\vdots & \vdots & \vdots & \ddots & \vdots \1 & u_{n-1} & u_{n-1}^2 & \cdots & u_{n-1}^{n-1}\end{pmatrix}.\]

Write the Determinant Formula for Vandermonde Matrix

The determinant of a Vandermonde matrix \( V \) is a well-established formula and is given by:\[\operatorname{det}(V) = \prod_{1 \leq j < i \leq n} (u_i - u_j).\]This product runs over all pairs \((i, j)\) with \( 1 \leq j < i \leq n \), i.e., for all pairs of distinct indices where \(i\) is larger than \(j\). This determinant formula reflects that the determinant is the product of differences of the entries in the first column.

Explain the Derivation Briefly

To prove this formula, observe that the determinant expands as an alternating product over permutations of the rows. Each term in the expansion corresponds to a product of entries from different rows, and permutation signs contribute to arrange terms combinatorially. The differences \((u_i - u_j)\) precisely factor the polynomial representing the determinant, reflecting the linear independence of the row entries when treated as polynomials in \(u_i\). Essentially, the calculation involves subtracting successive rows reduced by the row above them, until you achieve an upper triangular form with these differences on the diagonal.

Connect to Integral Domain

Since \( R \) is an integral domain, it follows that if \( \prod_{1 \leq j < i \leq n} (u_i - u_j) = 0 \), then at least one \( u_i - u_j = 0 \), which cannot happen as \( u_i \) must be distinct in an integral domain. Hence, the determinant is non-zero, reinforcing that the Vandermonde matrix is non-singular whenever \( u_i \) are distinct.

Key concepts

Determinant

The concept of a determinant is crucial when dealing with matrices, especially in determining properties like linear independence. In simple terms, the determinant is a value calculated from a square matrix. It tells us whether the matrix is invertible. For a Vandermonde matrix, the determinant takes a special form due to its particular structure.

• The determinant of a matrix gives an idea if the matrix has full rank.
• If the determinant is zero, the matrix is singular and doesn't have an inverse.
• If non-zero, the matrix is non-singular and invertible.

The determinant of a Vandermonde matrix is given by:\[\operatorname{det}(V) = \prod_{1 \leq j < i \leq n} (u_i - u_j)\]This formula shows that the determinant is a product of differences between the elements of the first row. In the context of being used over an integral domain, it implies the matrix becomes singular (i.e., determinant equals zero) if any two elements, such as \(u_i\) and \(u_j\), are equal. Otherwise, the determinant remains non-zero.

Integral Domain

An integral domain is a type of ring with specific properties that come into play in the world of algebra. Within an integral domain, the product of non-zero elements is non-zero, which means it has no zero divisors.

• This property ensures that if a product within the domain is zero, at least one factor involved is zero.
• Integral domains allow us to work with fractions as their elements can be inverted similar to integers in arithmetic.

In our given problem, the fact that \(R\) is an integral domain plays a crucial role. The non-zero determinant of the Vandermonde matrix relies upon there being no zero divisors. This means all values \(u_i\) are distinct—if they were not, the differences \(u_i - u_j\) would lead to a zero product, contradicting the integral domain's properties.

Linear Independence

Linear independence is a key concept used to analyze sets of vectors or rows in a matrix. Vectors are linearly independent if no vector in the set is a linear combination of others.

• If the vectors form the rows of an invertible matrix, they showcase full rank and are independent.
• In terms of a Vandermonde matrix, the specific form of the matrix entries demonstrates linear independence when all differences \(u_i - u_j\) are non-zero.

The non-zero determinant derived from the Vandermonde matrix confirms the linear independence of its rows. This is because the unique form of each row—represented by varying powers of \(u_i\)—ensures that none of them can be created by a combination of others. Thus, linear independence reflects the full rank and the invertibility of the matrix so long as \(u_i\) are distinct.