Detailed solutions must be provided for each question. Marks are not awarded for answers (even if correct); they are awarded for the arguments leading to the answers.

Use of words as well as mathematical symbols, and clear, concise, logical explanations of your arguments.

3. LetT : P11 :-Elbethemap:utlJeiiJIevlzlI11.I {a} Show that T is a linear map. (b) Shaw that T is SLlleCtill'B. [13} Use Pan {h} to conclude that dim{ker{T:|} = 11.. (d) Denep11p2,...1pn E keIIETjsnmLforeuhl ii Enfwehavepir] =11?lJra:ER.UsePaJt[e)mheIp}m mndudenat' = {p11p2,...1pn}isabssisforker[T]. 4 Let 5 3 1'5 A = 4 l 3 4 -'-1 ll Determine 1he fuiluw'ing: {a} The charameristie polynomial} of A. [b] The eigenvalues of A. {e} A basis for each eigenspaee of A. (d) The gemnetric mulplieities ufThe eigenvalues