Again not sure what to do here.
Subspace of R3 spanned by the vectors u1 = (1,1,1) and u2 = (2, 0, -1) is a plane passing through the origin. Express w = (1, 2, 3) in the form w = w1 + w2 where w1 lies in the plane and w2 is perpendicular to the plane.
Hi. I did it this way, which is a bit tedious but at least I understand what I'm doing.
First find the equation of the plane. Any point (x,y,z) on it is of the form , . So
whence eliminating a and b gives .
Now let where and .
Note that and lie in the plane. So take the dot product of w with and , noting that :
Solving gives ,