An orthogonal projection onto a subspace is defined as 
where 
is a matrix made up of the vectors that form a basis for the subspace. Think of an othogonal projection like this... it is the linear transformation that projects a "shadow" of a vector onto a subspace as if a light were held exactly at 
to the subspace and above the vector.
where is a matrix made up of the vectors that form a basis for the subspace. Think of an othogonal projection like this... it is the linear transformation that projects a "shadow" of a vector onto a subspace as if a light were held exactly at to the subspace and above the vector. Excercise 7-4.
Find the orthogonal projection onto the plane 
in 
.
in . 
and we have 1 equation, we know that two of the variables must be free parameters. Thus if we set 
and 
we can find the basis for the plane: 
.
and 
are free vectors and thus form the basis.
where 
Doing the appropriate calculations yields the following for 
: