1) let L be a linear operator over a finite or infinite dim complex vector space
ID: 2961733 • Letter: 1
Question
1) let L be a linear operator over a finite or infinite dim complex vector space V, and let U be a finite dim subspace of V with dim(U) greater than or equal to 1 that is invariant under L. Prove that L has at least one eigenvector u in U.
2) Let L be a linear operator over a finite dimensional vector space V. Let x be in V.
a) prove that there exists a finite m such that the set {x , Lx , ... , L^(m-1)x , L^(m)x} is dependent.
b) Now let m be such that {x , Lx , ... , L^(m-1)x , L^(m)x} is DEPENDENT and{x , Lx , ... , L^(m-1)x} is NOT dependent.
Prove that S = span(x , Lx , ... , L^(m-1)x) is an invariant subspace of L.
(note taht therefore S contains at least one eigenvector of L)
Explanation / Answer
(1)Let V be the vector space R^3, with x,y,z as axes, let U be the vector space consisting of the x-y plane. Let L be the linear operator rotating V by 90 degrees around the z axis. Clearly, L acts on any vector in U other than the zero vector by changing it's direction by 90 degrees. Therefore L has no eigenvector in U.
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