Test paper 4
Time allowed : 3 hours
(Allocate 20 marks for each question)
Let F be the vector space of infinitely differentiate complex functions
and let Pn be the subspace of complex polynomial functions of degree
less than n. For every A € C define Pn,\ = {eXzp \ p G Pn}. Show that
Pn,\ is a subspace of F and that
is a basis of Pn<\. If D denotes the differentiation map prove that
(1) Dn(e~Xzf) = e~Xz(D - Aid)"/;
(2) i> n , A =Ker(D-Aid)»;
(3) Pn,A is .D-invariant;
(4) B is a cyclic basis for D — A id.
If Dn,\ denotes the restriction of D to Pn,\ find the characteristic
polynomial of Dn,A- li fi ^ X show, by considering the characteristic
polynomial of Dn,\ + (/x — A) id, that (2?n,A — /J>id)n is invertible.
Given A = , use the Cayley-Hamilton theorem and euclidean
[c d\
division to show that every positive power of A can be written in the
form
An =
Time allowed : 3 hours
(Allocate 20 marks for each question)
Let F be the vector space of infinitely differentiate complex functions
and let Pn be the subspace of complex polynomial functions of degree
less than n. For every A € C define Pn,\ = {eXzp \ p G Pn}. Show that
Pn,\ is a subspace of F and that
is a basis of Pn<\. If D denotes the differentiation map prove that
(1) Dn(e~Xzf) = e~Xz(D - Aid)"/;
(2) i> n , A =Ker(D-Aid)»;
(3) Pn,A is .D-invariant;
(4) B is a cyclic basis for D — A id.
If Dn,\ denotes the restriction of D to Pn,\ find the characteristic
polynomial of Dn,A- li fi ^ X show, by considering the characteristic
polynomial of Dn,\ + (/x — A) id, that (2?n,A — /J>id)n is invertible.
Given A = , use the Cayley-Hamilton theorem and euclidean
[c d\
division to show that every positive power of A can be written in the
form
An =
