Is Vector w in the Image of Matrix A?

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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?
 
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yeland404 said:
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??
 
micromass said:
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)
 
yeland404 said:
so vector w is in I am ( A)

Yes, and what does it mean that w is in im(A)?
 
micromass said:
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
 
yeland404 said:
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??
 
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