5.4 Inner Product Spaces N Extends the Idea of Scalar Product in Other Spaces Besides R , and the Same for Measuring Angles Between Vector Spaces

5.4 Inner Product Spaces n Extends the idea of scalar product in other spaces besides R , and the same for measuring angles between vector spaces.

1. an inner product on a vector space V is an operation on V that assigns to each pair of n vectors x and y a scalar hx, yi (just like in R ), and it satisﬁes the following conditions: (I) hx, xi ≥ 0 with hx, xi = 0 ⇐⇒ x = 0 (II) hx, yi = hy, xi (commutative) (III) hαx + βy, zi = αhx, zi + βhy, zi (like the linear transformation), where α, β are n constants in R Here are particular examples:

n T • In R we have hx, yi = x y

m×n Pm Pn • In R we have hA, Bi = i=1 j=1 aijbij 2×3 P2 for example, if A, B ∈ R , then hA, Bi = i=1(ai1bi1 + ai2bi2 + ai3bi3) = (a11b11 + a12b12 + a13b13) + (a21b21 + a22b22 + a23b23)

R b • In C[a, b] we have hf, gi = a f(x)g(x)dx

Pn • In Pn we have hp, qi = i=1 p(xi)q(xi), where xi (1 ≤ i ≤ n) are n distinct real numbers.

2. a space with an inner product is called an inner product space (IPS).

1 3. Properties of inner products in an IPS:

• orthogonal vectors: u ⊥ v ⇐⇒ hu, vi = 0 • norm: ||x|| = phx, xi hx,yi • vector projection: let x, y ∈ X, then the vector projection of x on y is p = ||y||2 y. hx,yi y The length of this projection is α = ||y|| (so then p = α ||y|| ) • Cauchy Schwarz Ineq: |hu, vi| ≤ ||u|| · ||v||

4. Note that

|hu,vi| |hu, vi| ≤ ||u|| · ||v|| ⇐⇒ −1 ≤ ||u||·||v|| ≤ 1,

|hu,vi| so the quantity ||u||·||v|| must be the cos θ, for some θ (0 ≤ θ ≤ π/2). So we can deﬁne the angle between two vectors in any space.

5. A vector space is called normed linear space if to each vector v, w in V we associate a real number called norm, satisfying the following:

(I) ||v|| ≥ 0 with ||v|| = 0 ⇐⇒ v = 0 (II) ||αv|| = |α| · ||v||, where α is a scalar (III) ||v + w|| ≤ ||v|| + ||w||

6. the norm we have for the inner product matches the general norm (the one that gives the length of the vector), however there are other norms (some of the properties will not hold with these norms)

Pn T • ||x||1 = i=1 |xi|, where the vector x = (x1, x2, . . . , xn) –how the entire vector behaves globally T • ||x||∞ = max |xi|, where the vector x = (x1, x2, . . . , xn) –says what is the 1≤i≤n largest component of the vector Pn p 1/p T • ||x||p = ( i=1 |xi| ) , where the vector x = (x1, x2, . . . , xn) . The common Pn 2 1/2 p one is the 2-norm (p = 2): ||x||2 = ( i=1 |xi| ) = hx, xi (the only norm derived from inner product). • Frobenius Norm (2-norm for matrices, with a particular inner product): Let m×n p qPm Pn 2 A ∈ R . Then ||A||F = hA, Ai = i=1 j=1 aij

7. the distance between vectors x and y in a normed linear space is ||y − x||.