Chapter 6 – Key Concepts

Question 1

What is the inner product of two vectors u and v?

Answer 1

If u = [u₁ … u_n] and v = [v₁ … v_n], then their inner product is given as:

u^Tv = u⋅v = u_{as row matrix} v_{as col. matrix} = u₁v₁ + … + u_nv_n

Question 2

Let u, v, and w be vectors in Rⁿ, and let c be a scalar. Evaluate the following:

a. u⋅v = ? (How do they commute?)
b. (u+v)⋅w = ? (How do they distribute?)
c. (cu)⋅v = (How do they associate?)
d. u⋅u (≥ or ≤) 0? u⋅u = 0 ⇔ u = ?

Answer 2

a. u⋅v = v⋅u
b. (u+v)⋅w = u⋅w+v⋅w
c. (cu)⋅v = c(u⋅v) = u⋅(cv)
d. u⋅u ≥ 0, u⋅u = 0 ⇔ u = 0