\f\f\f\f\f\fProbII-I 1. Section 4.3 Problem Hal{e} in [FIS]. No justication is required. 2. Section 4.3 Problem ll] in [FIS]. 3. Prove the following result that was discussed and used in class: suppose E is the elementary matrix obtained from In by the row operation R, In ii E. Prove that for all A e MMgn] one has A i EA, Le. left multiplication of A by E implements the row operation that created E from In. all. Prove that if A,B IE MrxnR] are similar matrices, then deth] 2 deth}. 5. Section 5.1 Problem 2[a},[c} in [PIS]. 3. Let T be a linear operator on a nite-dimensional vector space V. [a] Show that T is invertible if and only.F if ll is not an eigenvalue of T. {b} If T is invertible, show that J. is an eigenvalue of T if and onlyr if Ill1 is an eigenvalue of T\". 7'. Suppose S : V } V and T : V 3. V are linear operators, with V finite dimensional. Suppose u E V is an eigenvector of S with corresponding eigenvalue as... and u is an eigenvector of T, with mrresponding eigen value AT. [a] Prove that v is an eigenvector of ST. [b] IGive a formula for the corresponding eigenvalue. 3. Section 5.1 Problem 3[a:l,[b] in [PIS]. 9. Section 5.1 Problem L1[b] [e],[h} in [FIS]. \f\f