Let us give here another try at dening an adjoint operator.
Theorem 0.1. Let V be a nite-dimensional inner product space. If T : V V is linear
then there exists a unique linear transformation T : V V such that for all v, u V ,
T (v ), u = v , T (u) .
(1)
SOLUTION: ASSIGNMENT 5
4.1.3 Is the subset of P2 given below a subspace? Find a basis if it is.
cfw_ p(t) : p (1) = p(2)
(p is the derivative).
S. Let S = cfw_ p(t) : p (1) = p(2). For any p1 , p2 S , k1 , k2 R, by
(k1 p1 + k2 p2 ) |t=1 = k1 p1 (1) + k2 p
MAT217 Spring 2013
Final exam Solutions
1. Let T : V V be linear with V a vector space over F. If v1 , . . . , vk are
eigenvectors of T for eigenvalues 1 , . . . , k respectively, with i = j for
i = j , show that cfw_v1 , . . . , vk is linearly independe
Problem 1. Let V be a nite dimensional vector space over a eld F. A subspace W of V
is called proper if W = V . Show that there exists a maximal proper subspace of V . That is,
show that that there is a proper subspace W of V such that if W is another pro
MAT217 Midterm Solution
March 9, 2011 (Wed)
1. (a) Let dim V = n, dim W1 = m1 , and dim W2 = m2 . From the dimension
theorem, we have
dim (W1 W2 ) = dim W1 + dim W2 dim (W1 + W2 )
= m1 + m2 dim (W1 + W2 )
m1 + m2 dim V
= m1 + m2 n 1 .
Therefore, W1 W2 mu
Uniqueness of nilpotent form
Let U : V V be linear and nilpotent. Here V is a nite-dimensional vector space over
a eld F . Recall the denition of a chain of length l.
Denition 0.1. A chain of length l is a set of vectors
cfw_v, U (v ), U 2 (v ), . . . , U
Problem: Let V be an n-dimensional F-vector space and k n. The purpose
of this problem is to show that
dim(Altk (V ) =
n
,
k
by completing the following steps:
1. Let W be a subspace of V and let B = (v1 , . . . , vn ) be a basis for V such
that (v1 , . .
MAT217 Final Solutions
Spring 2012
1. Let V and W be nite-dimensional vector spaces over a eld F with T, U
L(V, W ). Prove that
rank(T + U ) rank(T ) + rank(U ) .
Solution. We will rst show that
R(T + U ) R(T ) + R(U ) .
Let w R(T + U ). Then there exist