Math Problem Statement

Let a,b∈R and let u , v , w∈Rn , where v≠0 . Show that

projv(au+bw)=a(projvu)+b(projvw)

Solution

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Math Problem Analysis

Mathematical Concepts

Vector Projections
Linear Algebra
Dot Product
Vector Spaces

Formulas

Projection of a vector u onto v: projv(u) = (u ⋅ v / v ⋅ v) v
Linear property of dot product: (a*u + b*w) ⋅ v = a*(u ⋅ v) + b*(w ⋅ v)

Theorems

Linearity of Projections
Properties of Dot Product

Suitable Grade Level

Undergraduate (Linear Algebra)