Calculus Linear Transformation

Question

Fix x ̄ = (x1,...,xn) ∈ Rn. Define f : Rn → R such that f(y ̄) = ⟨y ̄,x ̄⟩. Show f is a linear transformation.

Answer 1

Let vectors x, and y be as given. Let f be as given. Then let v also be a vector in Rⁿ. Then,

f(y + v) = <y + v, x> = <y, x> + <v, x> = f(y) + f(v), satisfying the first condition of linear transformation.

Now let k be a scalar in R. We have that f(ky) = <ky, x> = k<y, x> = kf(y), satisfying the second condition.

Both conditions for having a linear transformation has been satisfied for f.