Calculus Orthonormal

Question

Let y = -5, -1, 3, -1

V1 = 1, -4, 2, -1

V2 = 2, 3, 4, -2

Compute the distance d from y to the subspace of R4 spanned by V1 and V2

Answer 1

The subspace spanned by two linearly independent vectors is a plane, so we need to find the distance between a point and a plane. The overall idea is to project the point onto the plane (by projecting onto each of the vectors) and then find the distance between the point and the projected point. I will skip most of the tedious details of calculation, but highlight the overall steps.

Project y onto V1:

We'll call this vector a. a=3/11*V1

Project y onto V2:

We'll call this vector b. b=1/33*V2

Project y onto the plane spanned by V1 and V2:

Since V1 and V2 are orthogonal (the dot product of V1 and V2 is 0), this is just a+b! We can call this vector c. c=a+b=(1/3,-1,2/3,-1/3)

Find the distance between y and c:

y-c=(-16/3,0,7/3,-2/3)

|y-c|=\sqrt{103/3}