Exercise 31 — Linear Algebra

Verify the Cauchy-Schwarz inequality by first proving Lagrange’s identity:

\left(\sum_{1\leq j\leq n}a_{j}b_{j}\right)^{2}=\left(\sum_{1\leq j\leq n}a_{j}^{2}\right)\left(\sum_{1\leq j\leq n}b_{j}^{2}\right)-\sum_{1\leq k<j\leq n}(a_{k}b_{j}-a_{j}b_{k})^{2}

and then noting that the final term is positive. This result is an improvement over Cauchy-Schwarz because it gives a formula for the difference between the two sides. Interpret that difference in $\mathbb{R}^{2}$ .

Hefferon, Linear Algebra, Linear Systems, Length and Angle Measures, Exercise 31

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