MA 2101 S (SEMESTER I 2023/2024) — MIDTERM TEST
19 SEPTEMBER 2023
Throughout, let F be a field of arbitrary characteristic.
For any m, n ∈ N , let Mm×n (F ) denote the vector space of all m×n -matrices with entries in F .
For any F -vector space V , let L(V ) denote the algebra of all F -linear operators on V .
1.
Let p, q, r, s ∈ F [x] be four polynomials over F of degree 6 3 . Prove or disprove each of
the following statements:
(a) if p(1) = q(1) = r(1) = s(1) = 0 , then p, q, r, s are linearly dependent.
(b) if p(0) = q(0) = r(0) = s(0) = 1 , then p, q, r, s are linearly dependent.
2.
Let V be a finite dimensional F -vector space, and let A and B be subspaces of V
satisfying A+B = V (i.e. every vector in V is a sum of an element of A with an element
of B ). Consider the set
S := { f ∈ L(V ) : f (A) ⊆ A and f (B) ⊆ B }.
(a) Show that S is a subpace of L(V ) .
(b) Determine dim(S) in terms of dim(V ) , dim(A) and dim(B) .
3.
Let V be a finite dimensional vector space over F , and let T ∈ L(V ) be a linear operator
on V . Show that
rank(T ) + rank(T 3 ) > 2 rank(T 2 )
4.
in N .
Let m, n ∈ N>0 be positive natural numbers, and let A, B ∈ Mm×n (F ) be given matrices.
Consider the linear map
T : Mn×m (F )
>
Mm×n (F )
given by
T (X) := A X B.
Show that if m 6= n , then T is not invertible.
5.
Let n ∈ N>0 , and let S, T ∈ Mn×n (F ) be square matrices such that
Sn = 0 = T n
but
S n−1 6= 0 6= T n−1
in Mn×n (F ) .
Show that S and T are similar matrices — that is to say, there exists an invertible matrix
A ∈ GLn (F ) such that S = A T A−1 .