Question

Let \( T, S : {R}^4 \to {R}^4 \) be two non-zero, non-identity \( {R} \)-linear transformations. Assume \( T^2 = T \). Which of the following is/are true?

• A. \( T \) is necessarily invertible
• B. \( T \) and \( S \) are similar if \( S^2 = S { and } {Rank}(T) = {Rank}(S) \)
• C. \( T \) and \( S \) are similar if \( S \) has only 0 and 1 as eigenvalues
• D. \( T \) is necessarily diagonalizable

Hint:

For linear transformations, check eigenvalues and ranks to determine properties like similarity and diagonalizability.

Answer 1

The Correct Option is B

Step 1: Analyzing \( T^2 = T \). This implies that \( T \) is idempotent. An idempotent operator is not necessarily invertible. 

Step 2: Similarity of \( T \) and \( S \). If \( S^2 = S \) and \( {Rank}(T) = {Rank}(S) \), \( T \) and \( S \) are similar because they represent the same type of projection. 

Step 3: Diagonalizability of \( T \). Idempotent operators are diagonalizable, with eigenvalues 0 and 1. 

Step 4: Conclusion. The correct answers are \( {(2), (4)} \).