mat224_final_review - MAT224 Final Review Session A brief...

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MAT224 Final Review SessionA brief summary of material since Test 2Asif ZamanApril 14, 2015Below is a very brief summary of the main concepts covered during the Tuesday April 14 review session.For more details, please see your lecture notes or the textbook.Also, note that the abbreviation “TP” is short for “Tutorial Problems”.1Eigenvectors & EigenvaluesLetVbe a finite dimensional vector space over a fieldF, and letT:VVbe a linear map.Recall:λFis an eigenvalue ofT(defn)⇐⇒ ∃x6= 0 such thatT(x) =λx.(thm)⇐⇒λis a root of thecharacteristic polynomialp(t) = det(T-tI).Recall:xVis aneigenvector ofTwith eigenvalueλF(defn)⇐⇒x6= 0 andT(x) =λx.(thm)⇐⇒x6= 0 andxKer(T-λI).Recall:theλ-eigenspace ofTisEλ={xV|T(x) =λx}.Facts:Eλis a subspace ofV1dimEλmλwheremλis the multiplicity of the rootλinp(t).The setS=[eigenvaluesλ{eigenbasis forEλ}is linearly independent.(Cayley-Hamilton) Ifp(t) = det(T-tI) has dimVroots inF, thenp(T) =0, the zero map.Some Examples:TP6 #2, 42DiagonalizationLetVbe a vector space of dimensionn= dimV

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