Linear Transformations and Image of a Matrix

[Ekaterina]

Homework Statement

Consider a 2x2 matrix A with A²=A.
If vector w is in the image of A, what is the relationship between w and Aw?

Homework Equations

Linear transformation T(x)=Ax
Image of a matrix is the span of its column vectors

The Attempt at a Solution

I know that vector w is one of the column vectors of A, seeing as it is in the image of A.
At first, I tried to work out Aw on paper by substituting [x₁ x₂] for the second column vector in A, since it isn't specified.
I got the following four equations:
w1 = w1^2+x1w2
w2=w1w2+x2w2
x1=w1x1+x1x2
x2=x1w2+w2^2
However, I couldn't get anything from there.

Now, I'm trying to think of it more conceptually (I believe the point of the problem is to think about it more abstractly), but I'm not sure which direction to head in. All I know is that Ax can equal w, so Aw may also equal w, but the problem doesn't say that w IS the image of A, just that it is IN the image of A.

Any suggestions on how to think about this?

 

[Dick]

Homework Statement

Consider a 2x2 matrix A with A²=A.
If vector w is in the image of A, what is the relationship between w and Aw?

Homework Equations

Linear transformation T(x)=Ax
Image of a matrix is the span of its column vectors

The Attempt at a Solution

I know that vector w is one of the column vectors of A, seeing as it is in the image of A.
At first, I tried to work out Aw on paper by substituting [x₁ x₂] for the second column vector in A, since it isn't specified.
I got the following four equations:
w1 = w1^2+x1w2
w2=w1w2+x2w2
x1=w1x1+x1x2
x2=x1w2+w2^2
However, I couldn't get anything from there.

Now, I'm trying to think of it more conceptually (I believe the point of the problem is to think about it more abstractly), but I'm not sure which direction to head in. All I know is that Ax can equal w, so Aw may also equal w, but the problem doesn't say that w IS the image of A, just that it is IN the image of A.

Any suggestions on how to think about this?

Asking "what is the relationship" is a pretty ambiguous question. I'd think about it more conceptually this way: A(Aw)=A^2(w)=Aw. What is A(Aw-w)? What do those tell you conceptually? Here's a hint about what they might be fishing for, look up projection operator. That's my best guess.