LINEAR ALGEBRA
IMPA - 2016
INSTRUCTOR: EMANUEL CARNEIRO
Problem Set 2
Problem 11. .(i) State Zorns lemma (you do not have to prove
it).(ii) Using Zorns lemma prove the following: every infinite
dimensional vector
space V has a basis B = {i}iI .
Note: Recall that a basis B (which may contain an uncountable
number ofelements) is a set such that every vector v V may be
written uniquely asa finite linear combination of elements of
B.
Problem 12. For n 3, let (b0, b1, b2, . . . , bn1) = (1, 1, 1,
0, 0, 0, 0, 0, . . . , 0). LetCn = (ci,j)nn be the matrix defined
by ci,j = b(ji)modn. Find det(Cn).
Note: m mod n is the remainder on the division of m by n.
Problem 13. Let A be an n n matrix, and X be an n n matrix with
equalentries. Show that
det(A+X) det(AX) detA2.Problem 14. Let A and B be two n n
matrices with complex entries such thatAB +BA = 0 and det(A+B) = 0.
Show that det(A3 B3) = 0.Problem 15. Let S be the subspace of Rnn
(the space of n n real matrices)generated by the matrices of the
form AB BA, with A,B Rnn. Find thedimension of S. (Hint: Show that
tr(AB) = tr(BA), where tr denotes the trace).
Problem 16. Let A be an n n matrix with a1,j = ai,1 = 1 (for 1
i, j n),and ai+1,j+1 = ai,j + ai+1,j + ai,j+1 (for all i, j with 1
i, j < n). Then
A =
1 1 1 1 1 3 5 7 1 5 13 25 1 7 25 63 ...
......
.... . .
Compute det(A).
Date: 13 de janeiro de 2016.2000 Mathematics Subject
Classification. XX-XXX.Key words and phrases. XXX-XXX.
1
2 EMANUEL CARNEIRO
Problem 17. Let A be an nn real matrix and At its transpose.
Show that AtAand At have the same range.
Problem 18. Let R[x1, x2, . . . xn] be the polynomial ring over
the field R in the nvariables x1, x2, . . . , xn. LetA be the
nnmatrix whose i-th row is (1, xi, x2i , . . . , xn1i )for i = 1,
2, . . . , n. Show that
detA =i>j
(xi xj).
Problem 19. Let n be a positive integer and let P2n+1 be the
vector space of realpolynomials of degree at most 2n+ 1. Prove that
there exist unique real numbersc1, c2, c3, . . . cn such that 1
1p(x) dx = 2p(0) +
nk=1
ck(p(k) + p(k) 2p(0))
for all p P2n+1.Problem 20. A mouse, who initially occupies the
cage A, is trained to changecages every time he hears a bell. Every
time he hears the bell, the mouse choosesone of the tunnels
adjacent to his cage with equal probability and moves along
thattunnel. After 23 rings of the bell, what is the probability
that the mouse is in cageB?
A B C
D E F
IMPA - Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil
22460-320E-mail address: [email protected]
Problem Set 2Problem 11Problem 12Problem 13Problem 14Problem
15Problem 16Problem 17Problem 18Problem 19Problem 20
