Prove that T^2 = T0 IFF R(T) C N(T)

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In summary, the conversation discusses the proof that a linear transformation T from vector space V to itself is equal to the zero mapping T0 if and only if the range of T is contained in the null space of T. The conversation includes a question about whether this reasoning is correct and clarifies that R(T) refers to the range of T and N(T) refers to the null space of T. It is noted that while T may not necessarily be the zero mapping, T^2(x) will equal 0 for all x if and only if T(T(x)) equals 0. A counterexample is given to show that R(T) does not necessarily have to be zero and T is not just the zero map.
  • #1
eckiller
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Let T be a linear transformation from V to V. Prove that T^2 = T0 (with T0 the zero mapping) IFF R(T) C N(T). ( "is contained in" = 'C'. )

Comments:

It seems clear that T is also the zero transformation. IF this is wrong can someone give me a counterexample. If it is true, then R(T) = { 0 } and N(T) = {v | v in V}, and clearly R(T) C N(T). That would show half, and if this part is right, then I can finish the second half. But is my reasoning for this part correct?--it almost seems too easy.
 
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  • #2
R(T) is the range and N(T) is the null space, correct?

R(T) need not be zero, and T is certainly not just the zero map.

T^2(x) = 0 for all x if and only if T(T(x))=0, ie if and only if T(x) is in...
 
  • #3
consider the shift operator, on R^2, taking (1,0) to (0,1) and (0,1) to (0,0).
 

1. What does the equation T^2 = T0 IFF R(T) C N(T) mean?

The equation T^2 = T0 IFF R(T) C N(T) is a mathematical statement that relates the square of a linear transformation T and the composition of its range and null space. It essentially states that the square of T will equal the zero transformation if and only if the range of T is a subset of the null space of T.

2. How is this equation used in linear algebra?

This equation is often used to prove properties and relationships between the range and null space of a linear transformation. It can also be used to determine if a linear transformation is invertible or not.

3. Can you provide an example of a linear transformation where T^2 = T0 IFF R(T) C N(T)?

Yes, consider the linear transformation T: R^2 -> R^2 defined by T(x, y) = (0, 0). In this case, the range of T is the zero vector and the null space of T is the entire R^2 space. Thus, T^2 = T0 and R(T) C N(T) holds true.

4. What are the implications of T^2 = T0 IFF R(T) C N(T) for the invertibility of a linear transformation?

If T^2 = T0 IFF R(T) C N(T) holds true, then the linear transformation T is not invertible. This is because the range of T is contained within the null space of T, making it impossible for T to have an inverse.

5. How does this equation relate to the concept of eigenvalues and eigenvectors?

This equation is closely related to the concept of eigenvalues and eigenvectors. If T is a square matrix, then T^2 = T0 IFF R(T) C N(T) is equivalent to saying that the only eigenvalue of T is 0. Additionally, the eigenvectors corresponding to an eigenvalue of 0 will be in the null space of T.

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