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Two linear transformations

Source: Iran PPCE 2012-Linear Algebra exam-P3

February 16, 2012
vectorlinear algebramatrixlinear algebra unsolved

Problem Statement

Suppose that T,U:V⟶VT,U:V\longrightarrow V are two linear transformations on the vector space VV such that T+UT+U is an invertible transformation. Prove that
TU=UT=0⇔rank⁡T+rank⁡U=nTU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n.
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