T2-2013-09-17-M_Gabarito

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UFRJ
Instituto de Matemática
Disciplina: Algebra Linear II - MAE 125
Professor: Bruno, Cesar, Flavio, Luiz Carlos, Ma-
rio, Milton, Monique e Paulo
Data: 17 de setembro de 2013
Segundo Teste
1. Considere o conjunto de vetores em R2
S = {(x, y), x ∈ R e y ∈ R}, com as operações de
soma vetorial e multiplicação por escalar dadas por:
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2 − 1),
k(x, y) = (kx, ky).
Assinale a afirmativa VERDADEIRA:
(a) A soma definida acima possui elemento
neutro
(b) (0, 0) 6∈ S
(c) S é um subespaço vetorial de R2
(d) S é um espaço vetorial
2. Assinale a solução geral do sistema linear represen-
tado pela matriz aumentada
 1 3 1 10 1 3 1
0 0 0 0

(a) (6,−2, 1) + t(16,−6, 2)
(b) (6,−2, 2) + t(16,−6, 2)
(c) (−2, 1, 0) + t(8, 3, 0) + s(0, 0, 1)
(d) (−2, 1, 0)
3. Assinale o conjunto que não é um subespaço vetorial
de R3:
(a) {(x, y, z) ∈ R3|y = 4x+ 2}
(b) {(a, b, c) ∈ R3|4a+ 5b = 2c}
(c) {(x, y, z) ∈ R3|y = 4x− 5z}
(d) {(a, b, c) ∈ R3|b = 5c}
4. Sejam os vetores u = (2,−3, 2) e v = (−1, 2, 4) em
R3. A condição entre a, b, c para que (a, b, c) seja
uma combinação linear de u e v é dada por:
(a) 16a+ 10b− c = 0
(b) 2a− 3b+ 2c = 0
(c) a− b− 2c = 0
(d) 5a+ 3b+ c = 0
5. Seja S = {v1,v2, · · · ,vk} ⊂ Rn e seja
A =
[
v1 v2 · · · vk
]
a matriz cujas colunas são os vetores de S e seja
B =

v1
v2
· · ·
vk

a matriz cujas linhas são os vetores de S. Considere
as afirmativas:
I Se S é um conjunto gerador de Rn então escalo-
nando a matriz A não se obtém nenhuma linha
de zeros.
II Se S é um conjunto gerador de Rn então escalo-
nando a matriz B não se obtém nenhuma linha
de zeros.
(a) A afirmativa I é verdadeira e II é falsa
(b) A afirmativa I é verdadeira e II é verdadeira
(c) A afirmativa I é falsa e II é falsa
(d) A afirmativa I é falsa e II é verdadeira
6. Considere as seguintes bases de R3,
α = {(1, 1, 1), (1, 1, 0), (1, 0, 0)}
β = {(0, 0, 1), (1, 0, 1), (1, 1, 1)}
Dado um vetor v ∈ R3 sejam [v]α e [v]β as coorde-
nadas de v nas bases α e β respectivamente. Se
[v]α =
 11
1

calcule a soma das entradas de [v]β
(a) 1
(b) 3
(c) −3
(d) −4
7. Considere as afirmativas:
I O espaço vetorial U =〈
x2 + 1, x+ 1, x2 + x+ 1
〉
possui dimensão
3.
II O espaço vetorial V =〈[
1 1
1 1
]
,
[
1 0
0 1
]
,
[
0 1
1 0
]
,
[
1 0
1 1
]〉
possui
dimensão 4.
(a) A afirmativa I é verdadeira e a afirmativa
II é falsa
(b) A afirmativa I é verdadeira e a afirmativa II é
verdadeira
(c) A afirmativa I é falsa e a afirmativa II é falsa
(d) A afirmativa I é falsa e a afirmativa II é verda-
deira
Gabarito Pág. 1
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Gabarito Pág. 6
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Gabarito Pág. 7
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Gabarito Pág. 8
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Gabarito Pág. 9