Is Vector w in the Image of Matrix A?

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Homework Help Overview

The discussion revolves around a 2x2 matrix A with the property that A^2 = A. Participants are exploring the implications of a vector w being in the image of matrix A, as well as the relationships between w and the transformations involving A.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are examining the relationship between w and A*w, questioning whether A^2*w equals A*w, and discussing the implications of w being in the image of A. There are inquiries about expressing w in a specific form and the meaning of being in the image of A.

Discussion Status

The discussion is active, with participants seeking clarification on the concept of the image of a matrix and its implications. Some have suggested that w can be expressed in terms of the transformation Ax, while others are probing deeper into the definitions and relationships involved.

Contextual Notes

Participants are considering the implications of different ranks of matrix A, although specific details about these ranks have not been fully explored in the current discussion.

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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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