Exercise 6.1.11

Answers

Compute that

∥ x + y ∥^{2} + ∥ x - y ∥^{2}

= (∥ x ∥^{2} + ⟨ x, y ⟩ + ⟨ y, x ⟩ + ∥ y ∥^{2}) + (∥ x ∥^{2} - ⟨ x, y ⟩ - ⟨ y, x ⟩ + ∥ y ∥^{2})

= 2 ∥ x ∥^{2} + 2 ∥ y ∥^{2} .

This means that the sum of the square of the four edges of a parallelogram is the sum of the square of the two diagonals.

jephianlin

2011-06-27 00:00