1. ## Orthogonal projection

Question: Find matrices of orthogonal projections onto all 4 fundamental subspaces of
the matrix
A =
1 1 1
1 3 2
2 4 3
Note, that really you need only to compute 2 of the projections. If you pick an appropriate 2, the other 2 are easy to obtain from them.

2. Ok so I know the the four fundamental subspaces are Ker(A), Ker(A*), Range(A) and Range(A*), where * denotes the conjugate transposed matrix. So after calculating for example Ker(A), which has (1,1,-2)^T as a basis, how does one proceed to find the matrix of orthogonal projection?

I also know that after I've found the matrix of orthogonal projection of Ker(A) that I need everyhting that is orthogonal to this space to find Ran{A*), since Ker(A)= Ran(A*) _|_, where _|_ denotes that it's orthogonal.
I just need help with the actual calculations..

3. Originally Posted by faust2016
Ok so I know the the four fundamental subspaces are Ker(A), Ker(A*), Range(A) and Range(A*), where * denotes the conjugate transposed matrix. So after calculating for example Ker(A), which has (1,1,-2)^T as a basis, how does one proceed to find the matrix of orthogonal projection?
The projection of vector u onto vector v is $\frac{u\cdot v}{|v|^2}v$. Thus, the projection of (1, 0, 0)^T onto (1, 1, -2)^T is $\frac{1}{6}$(1, 1, -2)^T, the projection of (0, 1, 0)^T is the same, and the projection of (0, 0, 1)^T is $\frac{-2}{6}$(1, 1, -2)^T= $\frac{-1}{3}$(1, 1, -2)^T. Those are the columns of the projection matrix.

I also know that after I've found the matrix of orthogonal projection of Ker(A) that I need everyhting that is orthogonal to this space to find Ran{A*), since Ker(A)= Ran(A*) _|_, where _|_ denotes that it's orthogonal.
I just need help with the actual calculations..
If the kernel is spanned by (1, 1, -2) then its orthogonal complement is spanned by any two vectors that are perpendicular to that. It is easy to see that (1, -1, 0) is perpendicular to (1, 1, -2) and the cross product of them, (2, 2, 2) is perpendicular to both.

4. Thanks a lot!