Is Vector w in the Image of Matrix A?

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In summary, the conversation discusses a 2*2 matrix A with A^2=A and different scenarios for the rank of A. It also touches on the relationship between a vector w and A*w, and how w being in the image of A means that it can be written as Ax for some x. It is also mentioned that the image of the linear transformation Ax is the span of the column vectors in A.
  • #1
yeland404
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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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  • #2
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??
 
  • #3
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)
 
  • #4
yeland404 said:
so vector w is in I am ( A)

Yes, and what does it mean that w is in im(A)?
 
  • #5
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
 
  • #6
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??
 

FAQ: Is Vector w in the Image of Matrix A?

What is a 2*2 matrix?

A 2*2 matrix is a rectangular array of numbers arranged in rows and columns. It is denoted by A, and its dimensions are 2 rows by 2 columns.

What does A^2 mean in a matrix?

A^2 is an operation known as matrix multiplication. It means multiplying matrix A by itself. In other words, it is the product of A with itself.

How do you determine if a matrix satisfies the equation A^2 = A?

To determine if a matrix A satisfies the equation A^2 = A, you need to multiply A by itself and then compare the resulting matrix to A. If they are equal, then A satisfies the equation.

Can a matrix satisfy the equation A^2 = A without being a square matrix?

Yes, a matrix can satisfy the equation A^2 = A without being a square matrix. As long as the number of columns in A is equal to the number of rows in A^2, the equation will hold true.

What is the significance of a matrix satisfying the equation A^2 = A?

A matrix satisfying the equation A^2 = A is known as an idempotent matrix. This means that multiplying the matrix by itself does not change its value. It has applications in various fields such as engineering, economics, and computer science.

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