let V be the cloumn space of matrix A
1 -1
1 2
1 -1
then the orthogonal projection of
0
1
0
on V is ?
Notice that the column vectors of your matrix are orthogonal, and hence form a basis for the subspace W spanned by
So the projection of becomes the sum
of its projections on subspaces that are mutually orthogonal. That is, the sum of the projections on subspace spanned by first vector in matrix and the projection on the subspace spanned by second vector in matrix.
I hope you know how to compute the orthogonal projection of one vector onto another.
Otherwise, here is the formula.
The first becomes:
and the second
Thus,