Inner Product

Question 1

How do we define dot product in R

Answer 1

We define dot product in R by:

X.y = sum(XiYi) = X^T *Y

Question 2

What is dot product in C

Answer 2

Dot product in C is:

X.y = sum(X(bar)iYi) = X(dagger)Y where x(dagger) is complex conjugate transpose of X

Question 3

What are properties of dot product

Answer 3

Properties of dot product are:
Conjugate symmetric .e x.y = (y.x)bar
Linear in 2nd slot, I.e x.(y+Lz) = x.y + Lx.z
X.x >= 0 with equality iff x =0

Question 4

What is definition of inner product

Answer 4

Definition of inner product is:

A map VxV to F : (v,w) which has certain properties

Question 5

What are properties of inner product

Answer 5

Properties of inner product are:
Conjugate symmetric inner(w,v) = inner(v,w)bar
Linear in 2nd slot inner(u,v+w) = inner(u,v) + inner(u,w), inner(u,Lv) = L(u,v) if F=R then also linear in 1st slot
Positive definite inner(v,v) >= 0 with equality iff v = 0

Question 6

What is an inner product space U

Answer 6

Inner product space U is:

U <= V where there is an inner product on U

Question 7

When is a map phi conjugate linear (or anti-linear)

Answer 7

A map phi is conjugate linear (or anti-linear) when:

Phi(v+Lw) = phi(v) + L(bar) phi(w)

Question 8

What is a sesquilinear function

Answer 8

Sesquilinear function is a function that is conjugate linear in the 1st slot and linear in the 2nd (inner products are sesquilinear)

Question 9

What is norm of v in inner product space V and orthogonal

Answer 9

norm of v in inner product space V is:

Root(inner(v,v)) >= 0 and if =0 then orthogonal

Question 10

What is Cauchy-Schwarz inequality

Answer 10

Cauchy-Schwarz inequality is:
For v,w in V,
Mod(inner product(v,w)) <= mod(v).mod(w) with equality iff v=0 or w=Lv