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### HW05

Course: EE EE, Spring 2011
School: National Chiao Tung...
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Word Count: 413

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203000 EE Linear Algebra Homework #5 (Due December 25, 2009 BEFORE class) Note: Detailed derivations are required to obtain a full score for each problem. (Total 110%) 1. (6%+6%) Problem 2(f) and Problem 3(d) of Section 5.1. 2. (6%+4%+6%+6%) Let T be a linear operator on a nite-dimensional vector space V. (a) Let us dene the determinant of T as det([T] ) for any ordered basis for V. Show that the determinant of...

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203000 EE Linear Algebra Homework #5 (Due December 25, 2009 BEFORE class) Note: Detailed derivations are required to obtain a full score for each problem. (Total 110%) 1. (6%+6%) Problem 2(f) and Problem 3(d) of Section 5.1. 2. (6%+4%+6%+6%) Let T be a linear operator on a nite-dimensional vector space V. (a) Let us dene the determinant of T as det([T] ) for any ordered basis for V. Show that the determinant of T is independent of the choice of the ordered basis. That is, show that det([T] ) = det([T] ) for any two ordered basis and for V. (b) Prove that T is invertible if and only if zero is not an eigenvalue of T. (c) Prove that is an eigenvalue of T if and only if 1 is an eigenvalue of T1 . (d) Prove that the eigenspace of T corresponding to the eigenvalue is the same as the eigenspace of T1 corresponding to 1 . 3. (8%) Find the general solution to the following system of dierential equations x (t) = 2x(t) + y (t) y (t) = x(t) + 2y (t). 4. (8%+4%) Problem 17 of Section 5.2. 5. (6%+6%+8%) For each of the following linear operators T on a vector space V, test T for diagonalizability, if and T is diagonalizable, nd an ordered basis for V such that [T] is a diagonal matrix (a) Let V = R3 and T be dened by a1 3a2 T a2 = a1 a1 + 2a3 a3 (b) Let V = P2 (R) and T be dened by T(f (x)) = f (x) + f (x)(x + x2 ). 1 (c) Let V = M22 (R) and T is dened by T(A) = At . 6. (6%+8%) Let T be a linear operator on P3 (R) dened by T(f (x)) = f (x) + 2f (x). (a) Find an ordered basis for the T-cyclic subspace W generated by z = x2 . (b) Compute the characteristic polynomial of TW by using Theorem 5.22 and verify that it divides the characteristic polynomial f (t) of T. 7. (8%+4%) Let T be a linear operator on a vector space V , where dim(V ) = n, and let its characteristic polynomial be given by f (t) = (1)n tn + an1 tn1 + + a1 t + a0 . (a) Prove that T is invertible if and only if a0 = 0. (b) Prove that, if T is invertible, then T1 = (1/a0 )[(1)n Tn1 + an1 Tn2 + + a1 IV ]. 8. (10%) Let A Mnn (F ) be an n n matrix such that A= M N O P , where M , N , and P are square matrices and O is an all zero matrix. Show that det(A) = det(M )det(P ). 2
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Linear AlgebraHomework #4 Solution1.F irst we want to show that det for any elementary matrix E.If E is of type 1, interchanging any row (column) of identity matrix I2, then1 11 0 0 11 0 1 00 1 0 10 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0Hence we
National Chiao Tung University - EE - EE
EE 203000 Linear AlgebraHomework #4(Due December 4, 2009 BEFORE class)Note: Detailed derivations are required to obtain a full score for each problem. (Total 50%)1. (14%) Let : M22 (F ) F be a function with the following three properties.(i) is a lin
National Chiao Tung University - EE - EE
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National Chiao Tung University - EE - EE
&quot;=&gt; '.I. (~)Q ~ v + W Sf., Cfl v+ vJ iS ~ s\,(b~pac.e,;I ~ l: W , ~ + ~f: V -tLA)8lAeh th ~t ~+ ~-=Q'~ ~ ~-\1'.' W 1S ex SL.tbspo.c.e. , 'i ~- (-~l)&quot;G=;&quot; ': Wi:LA)SLtbspo.~ . v ~ W t~Y\(-\.) :. 0 C v+ W C\&gt;-'d- ~ WIS cfw_A~ '!.-t-Le.
National Chiao Tung University - EE - EE
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National Chiao Tung University - EE - EE
1. a0 + a0= a(0 + 0)by (VS7)= a0by (VS3)= a0 + 0by (VS3)by theorem 1.1 a0 = 02.(1) It is not a vector space, since the conditions (VS2) (VS3) (VS4) (VS8) do not hold.(VS2) (a1 , a2 ), (b1 , b2 ), (c1 , c2 ) V(a1 , a2 )+(b1 , b2 )+(c1 , c2 ) =
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