HOMEWORK 2
(Math 250 A, B)
a b 1. (a) Given A c d , show that: 2 (i) p A ( x) x Tr(A)x + det(A) (ii) The eigenvalues of A are 1, 2 (a d ) (a d ) 2 4bc . 2
(20 pts)
(b) Let A be an nxn invertible matrix. Then, show that: (i) All the
TEST 4
(Math 250 A, B)
(Take Home)
1. The Caley-Hamilton Theorem provides a method for computing powers of a matrix A: (20 pts) 2 e.g. Let A be a 2x2 matrix with characteristic equation x ax b 0 .
Then, by the Caley-Hamilton Theorem, we have:
A
TEST 3
(Math 250 A)
1. (a) Define what it means for a linear map T : V W to be an isomorphism.
(30 pts)
(b) Suppose that dimV = dimW. Then show that a linear map T : V W is an isomorphism if and only if it is one-to-one or onto.
(Hint: Use the D
TEST 2
(Math 250 A)
1. (a) Show that the following map is a linear map: T : R3 R T ( x, y, z ) 2 x 3 y 4 z
(20 pts)
(b) Is the following map linear?
T : R2 R2 T ( x, y ) ( x 2 , y 2 )
2. (a) Explain why there is a unique linear map T : R 2