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 ).
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(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 ).
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