MATH 535, Fall 2025
Homework #3
Due on Wednesday, September 17th, 2025.
Problem 1. Let V be a finite-dimensional space over a field F, with n = dim(V ) > 0.
(a) Prove that every basis in V defines an algebra isomorphism Hom(V, V ) → Matn×n (F). Note
that here you need to prove that (i) it is an algebra homomorphism, that is K ◦ L is mapped
into the product of the appropriate matrices (in the same order), and (ii) it is an isomorphism
of vector spaces. For the sake of brevity, you do not need to prove the linearity of this map.
(b) Assume that A1 is the matrix of an operator L ∈ End(V ) for a basis B1 , and A2 is a matrix
of the same L for another basis B2 . Describe the relationship of the matrices A1 and A2 to
each other. Hint: you may start by expressing the basis B1 in terms of the basis B2 using yet
another square matrix, let’s call that matrix C. The answer will involve the matrix C...
Problem 2. Assume that V and W are finite-dimensional vector spaces, and that L ∈ Hom(V, W ).
(a) Consider the following bases: B = {x1 , . . . , xm } in V , and C = {y1 , . . . , yn } in W , as well as
the corresponding dual bases B 0 = {f1 , . . . , fm } in V 0 and C 0 = {g1 , . . . , gn } in W 0 . Show that
if an n × m matrix A is the matrix representation for L with respect to the given bases in
V and W , then the transposed matrix AT is the matrix representation of L0 ∈ Hom(W 0 , V 0 )
with respect to the dual bases.
(b) Use the Fredholm alternative to prove that rank(L) = rank(L0 ), and thus to conclude that the
row rank of a matrix is equal to its column rank.
Problem 3. Suppose V is finite-dimensional. Let M ⊂ V be a subspace, and assume that there
exist x1 , . . . , xn ∈ V such that {x1 , . . . , xk } form a basis in M and the elements {xk+1 + M, . . . , xn +
M } form a basis in V /M . Prove that {x1 , . . . , xn } is a basis in V .
Problem 4. Let M ⊂ V be a subspace of a finite-dimensional vector space V . Prove that there
is a natural (canonical) isomorphism M o → (V /M )0 , i.e. an isomorphism that does not depend on
any other choice, such as a choice of bases for these spaces.
Problem 5. Let Li : Vi → Vi+1 be a sequence of linear maps of finite-dimensional vector spaces,
{0} → V1 → V2 → V3 → · · · Vn−1 → Vn → {0},
where V0 = Vn+1 = {0}. Assume that every composition of two consecutive operators Li vanishes,
i.e. for every i = 1, . . . , n, we have Li ◦ Li−1 = 0. Prove that the following identity holds:
n
n
X
X
Ker (Li )
i
i
(−1) dim Vi =
(−1) dim
.
Im (Li−1 )
i=1
i=1
Problem 6. Suppose p ∈ F[t] with p 6= 0. Define a subspace U ⊂ F[t], U = {pq : q ∈ F[t]}. Show
that dim F[t]/U = deg p and find a basis in dim F[t]/U .