Question

Math

Posted 7 months ago

`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.`

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Answer from Sia

Posted 7 months ago

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.

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