Problem 34 - Calculus

Question

The Schwar inequality is \(\left(\sum_{1}^{n} a_{i} b_{i}\right)^{2} \leqslant\left(\sum_{i}^{n} a_{i}^{2}\right)\left(\sum_{1}^{n} b_{i}^{2}\right)\) Compute both sides if \(a_{1}=2, a_{2}=3, b_{1}=1, b_{2}=4\). Then compute both sides for any \(a_{1}, a_{2}, b_{1}, b_{2} .\) The proof in Section 11.1 uses vectors.

Step-by-Step Solution

Answer

For given values, 196 \(\leq\) 221 verifies the inequality holds.

Step 1: Compute Left Side of Inequality with Given Values

We're given the Schwar inequality \[\left(\sum_{1}^{n} a_{i} b_{i}\right)^{2} \leqslant \left(\sum_{i}^{n} a_{i}^{2}\right)\left(\sum_{1}^{n} b_{i}^{2}\right)\]First, let's calculate \(\sum_{i=1}^{2} a_{i} b_{i}\) using the values \(a_1 = 2, a_2 = 3, b_1 = 1, b_2 = 4\):\[\sum_{i=1}^{2} a_{i} b_{i} = (2 \times 1) + (3 \times 4) = 2 + 12 = 14\]Now square the result: \[\left(\sum_{i=1}^{2} a_{i} b_{i}\right)^{2} = 14^{2} = 196\]

Step 2: Compute Right Side of Inequality with Given Values

Next, compute \(\left(\sum_{i=1}^{2} a_{i}^{2}\right)\left(\sum_{i=1}^{2} b_{i}^{2}\right)\):First, calculate \(\sum_{i=1}^{2} a_{i}^{2}:\)\[\sum_{i=1}^{2} a_{i}^{2} = 2^{2} + 3^{2} = 4 + 9 = 13\]Then, calculate \(\sum_{i=1}^{2} b_{i}^{2}:\)\[\sum_{i=1}^{2} b_{i}^{2} = 1^{2} + 4^{2} = 1 + 16 = 17\]Multiply these results:\[\left(\sum_{i=1}^{2} a_{i}^{2}\right)\left(\sum_{i=1}^{2} b_{i}^{2}\right) = 13 \times 17 = 221\]

Step 3: Verify the Inequality

Compare the results from Step 1 and 2:\[196 \leqslant 221\]The inequality holds with the given values.

Step 4: Expression for General Computation

For any \(a_1, a_2, b_1, b_2\), the inequality is computed as:\[\left(a_1b_1 + a_2b_2\right)^2 \leqslant \left(a_1^2 + a_2^2\right)\left(b_1^2 + b_2^2\right)\]This leads to:1. Calculate \(a_1b_1 + a_2b_2\), then square it.2. Calculate \(a_1^2 + a_2^2\) and \(b_1^2 + b_2^2\), then multiply the two results.3. Verify that the squared sum is less than or equal to the product of sums.

Key Concepts

Inequalities in Mathematics

Inequalities are an essential part of mathematics. They allow us to compare two expressions or quantities without knowing their exact values. The Cauchy-Schwarz inequality is one such important tool. It is frequently used in various fields including algebra, physics, and computer science. In a basic sense, this inequality helps us determine how "aligned" two sets of numbers or vectors are. The original exercise asked to work with the expression \(\left(\sum_{1}^{n} a_{i} b_{i}\right)^{2}\) compared to \(\left(\sum_{i}^{n} a_{i}^{2}\right)\left(\sum_{1}^{n} b_{i}^{2}\right)\). The underlying idea is that the inner product (left side) squared is not more than the product of the sums of squares (right side).

Thus, by checking if the inequality holds, we basically ensure that multiplying the elements and their respective sums align or "sit well" with each other numerically. In simpler terms, the inequality has to do with the way sums of products (or the mix of two quantities) usually give a smaller outcome than their mixed sums of squares.

Vector Mathematics

When dealing with numerical datasets or lists, vector representation is widely used. Vectors are essentially an extension of simple numbers. In our exercise, the numbers \(a_1, a_2, b_1, b_2\) can be thought of as components of two vectors, say **\(\mathbf{a} = [a_1, a_2]\)** and **\(\mathbf{b} = [b_1, b_2]\)**. Vectors can make complex problems more digestible. They serve as a robust way to simplify matrix operations and geometry calculations.

The Cauchy-Schwarz inequality is a hallmark feature of vector mathematics. It tells us about the angle between the two vectors. Specifically, the inequality indicates that the dot product of two vectors (like **\(\mathbf{a} \cdot \mathbf{b}\)**) can't exceed the product of their magnitudes. When equality holds, the vectors are parallel. This becomes incredibly significant when dealing with multiple dimensions or when trying to project vectors onto one another. Understanding this aids in seeing the relationship between different sequences or directions depicting vectors.

Mathematical Proofs

Proofs stand as a cornerstone of mathematics, providing the 'whys' and 'hows' that give meaning to our calculations. With the Cauchy-Schwarz inequality, the proof usually involves steps rooted in algebra and sometimes geometric interpretations. It rationalizes why the expression involves squares and why they hold true regardless of specific numeric values.

A typical proof might start with assuming two vectors, the inner product, and then utilizing algebraic manipulation to show that the left-hand side is always less than or equal to the right-hand side. A common technique involves squaring both sides to eliminate the square root, transforming it into a recognizable quadratic form. Proving inequalities strengthens the understanding of mathematical rigor, offering insight into how abstract principles consistently apply across varied instances. It involves not just substitution but logical sequences that show the inherent truth backed by principles like linearity and symmetry.