# Math Help - Linear projection problem

1. ## Linear projection problem

The linear projection f is set with (2,1) -> (2,1,1) (1,1) -> (1,-1,2). Find the matrix that fits this linear projection in relation to the standard base, his formula and his core (subspace of all domain elements that project in the zero of the codomain).

tnx

2. So we know that

$T(2e_1+e_2)=2e_1+e_2+e_3 \iff 2T(e_1)+T(e_2)=2e_1+e_2+e_3$

and

$T(e_1+e_2)=e_1-e_2+2e_3 \iff T(e_1)+T(e_2)=e_1-e_2+2e_3$

Now we have a system of equations for the transformation of the basis vectors

So if we subtract the 2nd equation from the first we get

$T(e_1)=e_1+2e_2-e_3$

subbing into either equation gives

$T(e_2)=-3e_2+3e_3$

Now these are the columns of the transformation matrix.