Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 2 / b / i

Table of contents
- Solution i

Solution

Use the Euclidean inner product on

R^{n}

, with

a = (a_{1}, \dots, a_{n})

and

b = (b_{1}, \dots, b_{n})

. If

b = 0

, both sides of the desired inequality are zero. Otherwise take the single unit vector

v = b /∥ b ∥

in part (a). Then

∥ a ∥^{2} \geq \frac{⟨ a , b ⟩ ^{2}}{∥ b ∥ ^{2}},

so

(i = 1 \sum n a_{i}^{2}) (i = 1 \sum n b_{i}^{2}) \geq (i = 1 \sum n a_{i} b_{i})^{2} .

This is the real Cauchy-Schwarz inequality. Equality holds when the two vectors are linearly dependent.

 Ancestors (12)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version