Collection of Maths Problems

Cauchy-Schwarz inequality

Task number: 4475

• Solution

  Choose $\boldsymbol u=(a_1,\ldots,a_5)^{\mathrm T}$ and $\boldsymbol v=(4{,}3,6{,}4,2)^{\mathrm T}$, then with respect to the standard inner product on $\mathbb R^5$ we get $\langle\boldsymbol u|\boldsymbol v\rangle=4a_1 + 3a_2 + 6a_3 + 4a_4 + 2a_5$, $||\boldsymbol v||=\sqrt{4^2+3^2+6^2+4^2+2^2}=9$ and $||\boldsymbol u||=\sqrt{a_1^2 + \cdots + a_5^2}$.

  Then, according to the Cauchy-Schwarz inequality: $4a_1 + 3a_2 + 6a_3 + 4a_4 + 2a_5 = \langle\boldsymbol u|\boldsymbol v\rangle \le |\langle\boldsymbol u|\boldsymbol v\rangle|\le ||\boldsymbol v||\cdot||\boldsymbol u||= 9 \sqrt{a_1^2 + \cdots + a_5^2}$.