Is Vector w in the Image of Matrix A?

[yeland404]

Homework Statement

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

Homework 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 I am (A) along ker(A)

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？

 

[micromass]

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？

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

 

[yeland404]

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

so vector w is in I am ( A)

 

[micromass]

so vector w is in I am ( A)

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

 

[yeland404]

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

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

 

[micromass]

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

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