Let V be a finite-dimensional real vector space and let P be a linear transformation of V such that P^2 = P. Which of the following must be true? I. P is invertible. II. P is diagonalizable. III. P is either the identity transformation or the zero transformation.

Q: Let V be a finite-dimensional real vector space and let P be a linear transformation of V such that P^2 = P. Which of the following must be true? I. P is invertible. II. P is diagonalizable. III. P is either the identity transformation or the zero transformation.

∻Solution by Steps∻ ‖ step 1 ⋮ Consider the property $ P^2 = P $ for a linear transformation. ‖ ‖ step 2 ⋮ If $ P $ is invertible, then $ P^{-1}P = I $, where $ I $ is the identity transformation. ‖ ‖ step 3 ⋮ Multiplying both sides of $ P^2 = P $ by $ P^{-1} $ would imply $ P = I $, which is not necessarily true for all $ P $. Therefore, $ P $ is not necessarily invertible. ‖ ‖ step 4 ⋮ A linear transformation $ P $ satisfying $ P^2 = P $ is called a projection. ‖ ‖ step 5 ⋮ Projections are diagonalizable if there exists a basis of $ V $ consisting of eigenvectors of $ P $. ‖ ‖ step 6 ⋮ The eigenvalues of $ P $ are 0 and 1, since for any eigenvector $ v $, $ P^2(v) = P(v) $ implies $ P(v) = \lambda v $ and $ \lambda^2 v = \lambda v $. ‖ ‖ step 7 ⋮ Therefore, $ P $ is diagonalizable. ‖ ‖ step 8 ⋮ If $ P $ is the zero transformation, then $ P(v) = 0 $ for all $ v \in V $, and if $ P $ is the identity transformation, then $ P(v) = v $ for all $ v \in V $. ‖ ‖ step 9 ⋮ Since $ P $ can be a projection onto a subspace of $ V $ other than $ \{0\} $ or $ V $ itself, $ P $ is not necessarily the identity or zero transformation. ‖ ∻Answer∻ ⚹ II. P is diagonalizable. ⚹ ∻Key Concept∻ ⚹Projections and Diagonalizability⚹ ∻Explanation∻ ⚹A linear transformation $ P $ satisfying $ P^2 = P $ is a projection and such transformations are diagonalizable, but they are not necessarily invertible nor must they be the identity or zero transformation.⚹◊What is the minimal polynomial of the linear transformation P in this context?⍭ Generate me a similar question◊

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∻Solution by Steps∻ ‖ step 1 ⋮ Consider the property $ P^2 = P $ for a linear transformation. ‖ ‖ step 2 ⋮ If $ P $ is invertible, then $ P^{-1}P = I $, where $ I $ is the identity transformation. ‖ ‖ step 3 ⋮ Multiplying both sides of $ P^2 = P $ by $ P^{-1} $ would imply $ P = I $, which is not necessarily true for all $ P $. Therefore, $ P $ is not necessarily invertible. ‖ ‖ step 4 ⋮ A linear transformation $ P $ satisfying $ P^2 = P $ is called a projection. ‖ ‖ step 5 ⋮ Projections are diagonalizable if there exists a basis of $ V $ consisting of eigenvectors of $ P $. ‖ ‖ step 6 ⋮ The eigenvalues of $ P $ are 0 and 1, since for any eigenvector $ v $, $ P^2(v) = P(v) $ implies $ P(v) = \lambda v $ and $ \lambda^2 v = \lambda v $. ‖ ‖ step 7 ⋮ Therefore, $ P $ is diagonalizable. ‖ ‖ step 8 ⋮ If $ P $ is the zero transformation, then $ P(v) = 0 $ for all $ v \in V $, and if $ P $ is the identity transformation, then $ P(v) = v $ for all $ v \in V $. ‖ ‖ step 9 ⋮ Since $ P $ can be a projection onto a subspace of $ V $ other than $ \{0\} $ or $ V $ itself, $ P $ is not necessarily the identity or zero transformation. ‖ ∻Answer∻ ⚹ II. P is diagonalizable. ⚹ ∻Key Concept∻ ⚹Projections and Diagonalizability⚹ ∻Explanation∻ ⚹A linear transformation $ P $ satisfying $ P^2 = P $ is a projection and such transformations are diagonalizable, but they are not necessarily invertible nor must they be the identity or zero transformation.⚹◊What is the minimal polynomial of the linear transformation P in this context?⍭ Generate me a similar question◊

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