Summary: LINEAR ALGEBRA (MATH 317H)
Assignment 18 - Inner products, Orthogonality
(1) Let ( . , . ) be the standard inner product on Cn
, and let . , . be another
inner product. Show that there is a self-adjoint invertible (n× n)-matrix A
x, y = (x, Ay).
Self-adjoint means A = A
(2) Let V be a real vector space, and let be a function which assigns to
each pair of vectors a real number such that the following conditions are
· Skew-Symmetry: (x, y) = -(y, x)
· Linearity: (x + z, y) = (x, y) + (z, y)
· Non-degeneracy: If (x, y) = 0 for all x V then y = 0.
In this case is called a symplectic form and V is called a symplectic vector