Understanding Vandermonde's Identity: Explanation and Proof

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In summary, Vandermonde's Identity is a mathematical theorem that expresses the determinant of a matrix using the products of its elements, named after French mathematician Alexandre-Théophile Vandermonde. The formula for Vandermonde's Identity is det(A) = ∏<sub>i<j</sub> (a<sub>j</sub> - a<sub>i</sub>), with many applications in linear algebra and combinatorics. It is related to Pascal's Triangle through the binomial theorem, and can be generalized to matrices of any size.
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Chromium
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Hey guys,

Could someone give me a clear explanation of what Vandermonde's Identity is? I'm looking at the proof in my book and I'm having a difficult time understanding this. Fortunately I understand the rest of the section (which covers Binomial theorem, Pascal's identity and triangle).

thanks
 
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Could you be more specific as to what you don't get? The proof is rather straightforward.
 
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Off-Topic: Every time someone post's a thread on this topic, I keep wondering why people talk about Harry-Potter-stuff on the maths forums. :biggrin:
 

FAQ: Understanding Vandermonde's Identity: Explanation and Proof

What is Vandermonde's Identity?

Vandermonde's Identity is a mathematical theorem that expresses the determinant of a matrix using the products of its elements. It is named after the French mathematician Alexandre-Théophile Vandermonde.

What is the formula for Vandermonde's Identity?

The formula for Vandermonde's Identity is det(A) = ∏i (aj - ai), where A is a square matrix of size n x n and ai are the elements of the first row of A.

What is the significance of Vandermonde's Identity?

Vandermonde's Identity has many applications in mathematics, particularly in linear algebra and combinatorics. It is useful in solving systems of linear equations, calculating determinants, and proving identities in combinatorics.

How is Vandermonde's Identity related to Pascal's Triangle?

Vandermonde's Identity can be used to prove the binomial theorem, which is the expansion of (a + b)^n. This expansion is equivalent to the nth row of Pascal's Triangle, and the coefficients in this expansion can be derived using Vandermonde's Identity.

Can Vandermonde's Identity be generalized to matrices of any size?

Yes, Vandermonde's Identity can be generalized to matrices of any size. The formula for the determinant of a general Vandermonde matrix is det(A) = ∏i (aj - ai)n-j+i-1, where A is a matrix of size n x m and ai are the elements of the first column of A.

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