Let y 3 3 3 3 and W Span1 0 1 0 1 1 1 0The distance from y

Let y =[3 3 3 3], and W = Span{[1 0 1 0], [-1 1 1 0]}.The distance from y to W is

Solution

First we need to find the vector in the span of W which is closest to y

Since the last entry in both the vectors is equal to zero, hence the minimum distance will be sqrt(9) or 3 units

Now we need to set the W vector such that the distance is minimized, so we would try to make atleast one row as 3, but we see that the below combination yields the minimum distance

W = 3[1 0 1 0] + [-1 1 1 0]

=>[2 1 4 0]

Distance = sqrt( (3-2)^2 + (3-1)^2 + (3-4)^2 + (3-0)^2 )

=> sqrt(1+4+9+1)

=>sqrt(15