# Thread: Some problem here :(

1. ## Can anyone please explain some Subspace problem here :(

Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
a. Find the standard matrix A of F
b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?

2. Originally Posted by pdnhan
Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
a. Find the standard matrix A of F
b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?
The vector normal to W is $\mathbf{n}=(1,1,1)$. Given a vector $\mathbf{v} = (x,y,z)$ in $\mathbb{R}^3$, the effect of F on $\mathbf{v}$ will be to add a multiple of $\mathbf{n}$ to $\mathbf{v}$ so as to take it to the subspace W. The condition for $\mathbf{v} + \lambda\mathbf{n} = (x+\lambda,y+\lambda,z+\lambda)$ to belong to W is $x+y+z+3\lambda = 0$. Solve that to see that $F(\mathbf{v}) = (\tfrac23x-\tfrac13y-\tfrac13z, -\tfrac13x+\tfrac23y-\tfrac13z, -\tfrac13x-\tfrac13y+\tfrac23z)$. From that, you should be able to write down the matrix A and verify that $A(A-I)=0$.

For the last part, if A is the matrix of the orthogonal projection onto a subspace, then I–A is the matrix of the orthogonal projection onto the (orthogonal) complementary subspace.

3. cheers man

4. hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?

5. Originally Posted by pdnhan
hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?
You tell us yours first, then I'll let you know whether I agree.