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JJ__
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- If T^2 = T, where T is a linear operator on a nonzero vector space V, does this imply that either T equals the identity operator on V or that T is the zero operator on V?
I can't think of a counterexample.
Because, as fresh_42 pointed out, the claim is NOT TRUE!Eclair_de_XII said:Why not start with the equality ##A^2=A## where ##A## is the matrix for the transformation ##T##, and prove the claim using the matrix algebra of transformations?
JJ__ said:Summary: If T^2 = T, where T is a linear operator on a nonzero vector space V, does this imply that either T equals the identity operator on V or that T is the zero operator on V?
I can't think of a counterexample.
And how does my idempotent example in post #2 fit into this scheme?StoneTemplePython said:... all idempotetent matrices are similar to this.
in reals, withfresh_42 said:And how does my idempotent example in post #2 fit into this scheme?
Because the ring of matrices is not an integral domain, so that ##A^2-A=0 \rightarrow A(A-I)=0 ## does not imply A=0 or A=I.Eclair_de_XII said:Why not start with the equality ##A^2=A## where ##A## is the matrix for the transformation ##T##, and prove the claim using the matrix algebra of transformations?
##A^2=A## is the definition of a projection.Eclair_de_XII said:Right, because ##A## can also represent the matrix of the projection transformation for some proper subspace ##U## of ##V##, and ##(A-I)## can represent the projection transformation for the complement of ##U##. In turn, the composition of these two linear transformations is just the zero transformation, I think?
A linear operator is a mathematical function that maps one vector space to another, while preserving the operations of addition and scalar multiplication.
This notation means that when the linear operator T is applied twice, it results in the same output as when it is applied once. In other words, the operator T is idempotent.
If T=I, it means that the linear operator T is the identity operator, which means it does not change the input vector. If T=0, it means that the linear operator T is the zero operator, which means it always outputs the zero vector.
Yes, there can be other solutions depending on the vector space V and the specific linear operator T. For example, if T is a projection operator, then T^2 = T for any nonzero projection operator.
This equation can be useful in simplifying calculations and understanding the behavior of the linear operator T. It can also help identify special properties of the operator, such as being an identity or zero operator.