Let \( T : {R}^4 \to {R}^4 \) be an \( {R} \)-linear transformation such that 1 and 2 are the only eigenvalues of \( T \). Suppose the dimensions of \({Kernel}(T - I_4)\) and \({Range}(T - 2I_4)\) are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of \( T \)?
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Solution and Explanation
Step 1: Eigenvalue properties and Jordan blocks. The eigenvalues of \( T \) are 1 and 2. The dimensions of the kernel and range provide information about the size of the Jordan blocks. - Dimension of \({Kernel}(T - I_4) = 1\): Indicates one Jordan block corresponding to \( \lambda = 1 \). - Dimension of \({Range}(T - 2I_4) = 2\): Indicates two Jordan blocks for \( \lambda = 2 \).
Step 2: Analyzing each option. - (1): This corresponds to one Jordan block for \( \lambda = 1 \) and two for \( \lambda = 2 \), consistent with the given conditions. - (2): This has more than one Jordan block for \( \lambda = 1 \), violating the kernel dimension condition. - (3): This includes a defective Jordan block for \( \lambda = 1 \), which is inconsistent with the kernel dimension condition. - (4): This corresponds to one Jordan block for \( \lambda = 1 \) and two for \( \lambda = 2 \), consistent with the conditions.
Step 3: Conclusion. The possible Jordan canonical forms are \( {(1) and (4)} \).
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