Need help with homework problems #3 and 5 Please answer thoroughly and show work! Thank you!

5 Vector Space Let V/ K be a vector space, 1) E V a nonzero vector and 8(1)) C [,(V, V) the set of linear transformations T : V e V for which 1) is an eigenvector: 5(1)) = {T e (V, V) : (3A 6 1mm) = m} Prove that 8(1)) is a vector space. Proof. 3 Composition Let V/K be a vector space7 let 37 T E (V7 V) and let B = {v17 . . . ,vn} C V. Assume that B is a basis of eigenvectors of both 5' and T. Prove that SOT = To 5'. Proof