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(c) I - D is a singular linear transformation on U.
(d) ForA a nonnegative integer, XI - D is singular.
(e) For A a nonnegative integer greater than 1, I - AD is nonsingular.
4. Let U = Span (sin x, cos x, sin 2x, cos 2x....
(a) Show that D is a nonsingular linear transformation on.
b) Determine the inverse of D.
c) Show that I + D I - D and I- D- are nonsingilar linear transformations
on U.
(d) Show that I + D2 is a singular linear transformation on U.
(e) Discuss the singularity or nonsingularity of I + D for an integer.
f) Discuss the singularity or nonsingulrity of I + ADI for an integer
5. Let A be a linear transformnation with x2 - x + 6 as minimal polynomial. Prove:
(a) A3= 19A - 301. (b) A4 = 65 A - 1141.
(c) Each polynomial in A of degre 2,.1,.. is equal to a polynonial in A of
degree 1.
(d) For n= 2, 3, 4,... A = sA + t,, where sn+ = S + t,,n = -S,
S., = 5, t,2 = -6. [Hint. Use induction.]
(e) The set of all real polynomials in A is a real vector space of dimension 2.
6. (a) Prove the rule T"4 = Tk 1 TIk'1 in (9-212). [Hint. It is sulcient to prove
Tk+l - TTL. By (9-201) in Section 9-20, we have (i) T = TT'- TT1k for
k > 0, 1 > 0. From these relations deduce the relations (for k > 0, 1 > 0):
(ii) Tk+T-k = T', (iii) T -k-Tk = T- (iv) T1 T1 kT
Show that (i)..... (iv) give the desired rule for all combinations of positive
and negative exponents.]
(b) Prove the rule (T)1' - 1kI in (9-212). [lInt. If 1 0 (J1k), - I -, and
we can apply the result of (a) if 1 0 write 1 = -, show that (I =
(T-k)m and apply the previous result if l = 0 verify the rule diuetly.]
7. Let g(x) be a minimal polynomial for - Prove:
(a) p(x)a = cg(x) is also a minimal polynoial for I povided that c 0.
(b) If p(x) is a minimal polynomial for 7 then p(x) g(x) for som 0. Hint.
p and g must have the same degree k, so that p(x)- d x +.,and
g(x) = c<k + Now cons ider (X) = p(A) - (dk/ C)g(x).]
9-23 EIGENVECTORS AND EIGENVALUES
Let T be a linear transformation on a vector space U. For some nonzero
vector u in U it may happen that
T(u) = Au
for an appropriate scalar N. When this happens, we call u an eigcnvetor of
T and we call an eigenvalue for T. We also say that u is an eigenvector
associated with the eigenvalue A. If u is an eigenvector associated with A,
