The norm

Question 1

norm

Answer 1

gievn a vector space V over R or C a norm on V is a function V-> scalars (R or C) satsifying:
‖λv‖ = λ‖v‖
‖ v+u‖ ≤ ‖v‖ + ‖u‖
‖v‖ ≥ 0 and = 0 iff v = 0

Question 2

norm and real inner product space

Answer 2

‖v‖ = √(v,v)

Question 3

unit vector

Answer 3

v is if ‖v‖ = 1

Question 4

orthogonal

Answer 4

say u,v are orthogonal if (u,v) = 0

Question 5

orthonormal basis for V

Answer 5

is a basis {u1…un} where (ui,uj) = 1 if i = j and = 0 if i≠j

Question 6

pythagoras theorem

Answer 6

if u,v orthogonal ‖u+v‖^2 = ‖u‖^2 + ‖v‖^2

Question 7

real cauchy schwartz inequality

Answer 7

(u,v)^2 ≤ (u,u)(v,v)
or
|(u,v)| ≤ ‖u‖‖v‖

Question 8

triangke inequality

Answer 8

‖u+v‖≤ ‖u‖ + ‖v‖

Question 9

angle between two vectors in terms of norm and inner product

Answer 9

(u,v) = ‖u‖ ‖v‖ cosθ

Question 10

norm induced by a hermitian inner product

Answer 10

letV be a hermitian inner product space then the norm ‖v‖ = √<v,v>

Question 11

hermitian norm

Answer 11

• unit vector same
• orthogonal same
• cauchy schwartz same but have |<u,v>|^2 ≤ ‖u‖^2 ‖v‖^2
  triangle inequality same

Question 12

angle between 2 vectors in compelx space

Answer 12

cant be defined but can still be orthogonal