# Thread: find the matrix for ProjL with respect to the basis

1. ## find the matrix for ProjL with respect to the basis

Let L be the line in R3 spanned by v1=[1 1 1]
(a)Find a basis {v1, v2} for the plane perpendicular to L.
(b) Let ProjL be the projection onto the line L. Find the matrix for ProjL with respect to the basis B.

Sol: We need to find 2 vectors that span the above plane, that is, we need to find the null space of the matrix. I got [1 1 1 0], which is already in row reduced echelon form. After doing some algebra, I got span([0 -1 1], [-1 0 1]). Therefore, the basis of R3 is {[0 -1 1], [-1 0 1]} right? But I don't get part (b). Help!

2. Pick two linearly-independent vectors w,v which each have zero dot product with [1,1,1]. Then (w,v) is a basis for the plane perpendicular to [1,1,1], and (w,v,[1,1,1]) is a basis for R3. Then projection onto the line L sends w and v to zero, and [1,1,1] to itself, which is all you need to know to write its matrix under that basis.