# A 2*2 matrix A with A^2= A

1. Sep 20, 2011

### yeland404

1. The problem statement, all variables and given/known data
a 2*2 matrix A with A^2= A
1)if vector w is in the image of A , what is the relationship between vector w and A*w

2. Relevant equations
2)what can say about A if rank(A)= 2 , what if rank(A)=0
3)if rank(A) = 1,show that the linear transformation T(x)=Ax is the projection onto im (A) along ker(A)

3. The attempt at a solution
to the 1), does the A^2*w= A w, and w is in the image of A, so the both w and Aw is in image A？

2. Sep 20, 2011

### micromass

What does it mean that w is in the image of A?? Can you write w=... in a special way??

3. Sep 20, 2011

### yeland404

so vector w is in im ( A)

4. Sep 20, 2011

### micromass

Yes, and what does it mean that w is in im(A)?

5. Sep 20, 2011

### yeland404

vector w belons to the image of matrix A, and the image of linear transformation Ax is the span of the column vector in A

6. Sep 20, 2011

### micromass

Can you prove that there exists an x such that w=Ax??