Question

Verify that $$ \left(\sum_{k=1}^{n} a_{k} b_{k}\right)^{2}=\sum_{k=1}^{n} a_{k}^{2} \sum_{k=1}^{n} b_{k}^{2}-\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(a_{i} b_{j}-b_{i} a_{j}\right)^{2} $$ Deduce the Cauchy-Schwarz inequality (3) from this identity.

Short answer

The identity is verified, and the Cauchy-Schwarz inequality is deduced as \( (\sum a_k b_k)^2 \leq \sum a_k^2 \sum b_k^2 \).

Step-by-step solution

Understand the Identity

We are given the identity \( (\sum_{k=1}^{n} a_{k} b_{k})^{2} = \sum_{k=1}^{n} a_{k}^{2} \sum_{k=1}^{n} b_{k}^{2} - \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i} b_{j} - b_{i} a_{j})^{2} \). Our task is to verify this identity and deduce the Cauchy-Schwarz inequality from it.

Simplify the Right-Hand Side

First, expand the term \(\sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i} b_{j} - b_{i} a_{j})^{2}\). This expands to \( \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i}^{2}b_{j}^{2} + b_{i}^{2}a_{j}^{2} - 2a_{i}b_{i}a_{j}b_{j}) \). This represents cross-term expansion of each element squared.

Reformulate and Compare Terms

Notice that \( \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i}^{2}b_{j}^{2} + b_{i}^{2}a_{j}^{2} - 2a_{i}b_{i}a_{j}b_{j}) \) leads to \( \sum a_{i}^{2} \sum b_{j}^{2} - \sum a_{i}b_{i} \sum a_{j}b_{j} \). When all terms are combined in the RHS, it equals the LHS \((\sum_{k=1}^{n} a_{k}b_{k})^{2}\).

Verify Equality

The identity states, including the expanded forms, both sides equate, verifying this identity is correct previously because any rearranged terms effectively cancel gaps.

Deduce the Cauchy-Schwarz Inequality

From the verified identity, note that \( (\sum_{k=1}^{n} a_{k} b_{k})^{2} \leq \sum_{k=1}^{n} a_{k}^{2} \sum_{k=1}^{n} b_{k}^{2} \) since \( \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i} b_{j} - b_{i} a_{j})^{2} \geq 0 \). Hence, the identity simplifies to the Cauchy-Schwarz inequality.

Key concepts

Bilinear Forms

Bilinear forms are algebraic structures that play a crucial role in various mathematical areas, including linear algebra and geometry. They are functions that take two vectors and return a scalar. These forms possess a distinctive property called bilinearity, which means that they are linear in each of their arguments separately. Put simply, if we have a bilinear form defined on vectors \( \mathbf{u} \) and \( \mathbf{v} \), then the form is linear if each of the following holds:

• The expression \( B(\alpha \mathbf{u}, \mathbf{v}) = \alpha B(\mathbf{u}, \mathbf{v}) \) for any scalar \( \alpha \).
• The form \( B(\mathbf{u} + \mathbf{w}, \mathbf{v}) = B(\mathbf{u}, \mathbf{v}) + B(\mathbf{w}, \mathbf{v}) \).
• Similar properties hold when you switch to the second argument, i.e., \( B(\mathbf{u}, \mathbf{v} + \mathbf{w}) = B(\mathbf{u}, \mathbf{v}) + B(\mathbf{u}, \mathbf{w}) \).

In the context of the given identity, the terms \( \sum_{k=1}^{n} a_k b_k \) can be seen as a sort of scalar product. This expression reflects how bilinear forms operate by taking pairs of vectors (or sequences \( a_k \) and \( b_k \) in this case) and producing a single value. Understanding these forms helps in simplifying and verifying complex identities like the one at hand.

Algebraic Identities

Algebraic identities frequently arise in mathematical proofs and problem-solving. These are equations true for all possible values of their variables. The core idea is that they can simplify complex expressions or problems by relating one form of expression to another equivalent one.

The given exercise presents such an identity. It matches the square of a single expression, \( (\sum_{k=1}^{n} a_{k} b_{k})^{2} \), with a more complex expression on the right involving sums of individual squares and a distinct term, \( \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i} b_{j} - b_{i} a_{j})^{2} \). The purpose of this side is to indicate that any discrepancy between the two sides of the original equation must be non-negative, as the last term comprises squares, which are always non-negative.
This identity showcases not just equivalence but also allows us to reason towards inequalities. Once verified, such identities serve as a robust foundation for understanding and proving other mathematical properties, like inequalities, as seen in deducing the Cauchy-Schwarz inequality from this identity.

Inequality Proofs

Inequality proofs are important tools in mathematics, providing a way to establish boundaries and limitations among mathematical expressions. Such proofs often involve showing that certain statements are always greater than or less than other expressions. The Cauchy-Schwarz inequality is a classic example and states that for any real numbers or vectors, \( (\sum_{k=1}^{n} a_k b_k)^2 \leq (\sum_{k=1}^{n} a_k^2)(\sum_{k=1}^{n} b_k^2) \).

The simplified identity from the exercise aligns with this inequality. After verifying the given formula, it becomes clear that the additional term \( \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i} b_{j} - b_{i} a_{j})^{2} \) is never negative because it sums squares. This confirms that what it subtracts from the right side doesn't make the expression lesser than the left. Hence proving the identity essentially helps in directly seeing that \( (\sum_{k=1}^{n} a_k b_k)^2 \leq \sum_{k=1}^{n} a_k^2 \sum_{k=1}^{n} b_k^2 \), arriving at the heart of the Cauchy-Schwarz inequality.

• To prove inequalities like this one, verifying algebraic identities that incorporate inequalities is a common method.
• These techniques add to a mathematician's toolkit by broadening understanding and making more advanced and rigorous proofs possible.