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evilpostingmong
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Homework Statement
Suppose S, T [tex]\in[/tex] the set of linear transformations
from V to V. Prove that ST and TS have the same eigenvalues.
Homework Equations
T=[tex]\lambda[/tex]I
The Attempt at a Solution
let v[tex]\in[/tex]V.
For TS
T(Sv)=T(I[tex]\lambda[/tex]v)=[tex]\lambda[/tex]T(Iv)
=[tex]\lambda[/tex]'[tex]\lambda[/tex]Iv.
For ST
S(Tv)=S([tex]\lambda[/tex]'v)=[tex]\lambda[/tex]'S(Iv)=
[tex]\lambda[/tex]'IS(v)=[tex]\lambda[/tex]'I[tex]\lambda[/tex]Iv
=[tex]\lambda[/tex]'[tex]\lambda[/tex]Iv.