LINEAR ALGEBRA
IMPA - 2016
INSTRUCTOR: EMANUEL CARNEIRO
Problem Set 1
Problem 1. Let T : R4 → R3 be the linear map:
T (x1 , x2 , x3 , x4 ) = (x1 + 2x2 − x3 − x4 , 2x1 + 4x2 + x3 + 10x4 , x1 + 2x2 + x3 + 7x4 ).
Find a basis for N (T ) and a basis for R(T ). What is the rank of T ?
Problem 2. Let P3 denote the real subspace of R[x] of polynomials of degree at
most 3. Let T : P3 → P3 denote the linear transformation
T (p(x)) = xp0 (x) − p(x).
Find a basis for N (T ) and a basis for R(T ). What is the rank of T ?
Problem 3. Let V be an n-dimensional vector space.
(i) Show that a proper subspace W of V is the intersection of all subspaces of
V of dimension n − 1 which contain W .
(ii) Let L(V ) be the vector space of all linear transformations of V to itself.
For x 6= 0 in V , compute the dimension of Lx (V ) = {T ∈ L(V ); T (x) = 0}.
Problem 4. Let T and S be linear transformations from Rn to Rm . The coincidence set for T and S is defined to be the set C(T, S) = {w ∈ Rm ; T (x) = w =
S(y), for some x, y ∈ Rn }. Let T and S be the linear transformations from R4 to
R3 represented by the 3 × 4 matrices:




1 2 0 −1
1 2 1 0
T =  5 4 1 0  and S =  −1 0 3 2  .
3 0 1 2
−1 4 11 6
Find a basis for C(T, S).
Problem 5. In the Euclidean space R4 consider the ellipsoid:
2x21 + 3x22 + 4x23 + 5x24 = 1.
Does there exist a 3-dimensional subspace of R4 which intersects the ellipsoid in a
sphere?
Date: 13 de janeiro de 2016.
2000 Mathematics Subject Classification. XX-XXX.
Key words and phrases. XXX-XXX.
1
2
EMANUEL CARNEIRO
Problem 6. Let p be a prime and F = Zp be the field with p elements. Let n ∈ N.
(i) What is the number of elements in the vector space F n ?
(ii) How many linear transformations T : F n → F n are there?
(iii) How many linear transformations T : F n → F n are surjective?
−4 18
Problem 7. Consider the matrix A =
.
−3 11
(i) Find an invertible matrix P such that P −1 AP is a diagonal matrix.
(ii) Find a closed formula for An , for n ∈ N, where An is the result of multiplying A by itself n times.
(iii) Consider the sequences of numbers {a0 , a1 , . . . } and {b0 , b1 , . . .} given by:
a0 = 1; b0 = 0; an+1 = −4an + 18bn ; bn+1 = −3an + 11bn . Find a closed
formula for an and bn .
Problem 8. Let A and B be n×n invertible matrices. Assume that (AB)k = Ak B k
for three consecutive natural numbers k. Show that AB = BA.
Problem 9. Let A and B be n × n matrices. Assume A is invertible and that for
all positive integers k we have (A + B)k = Ak + B k . Prove that B = 0.
Problem 10. Let A and X be matrices n × n and n × 1, respectively, with all
entries real and strictly positive. Assume that A2 X = X. Show that AX = X.
IMPA - Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil 22460-320
E-mail address: carneiro@impa.br