Teste1/19-1 de Álgebra Linear II - UFRJ

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Universidade Federal do Rio de Janeiro
Instituto de Matemática
Disciplina: Álgebra Linear II
Professor: Alejandro Cabrera, Bruno Costa, Flá-
vio Dickstein, Mário de Oliveira, Milton Ramirez,
Monique Carmona, Marianty, Nilson Bernardes
Data: 26 de Março de 2019
Teste
1. Assinale o conjunto-solução do sistema{
x+ y + z = 3
x− y + z = −5
(a) (−1, 4, 0) + t(−1, 0, 1), t ∈ R
(b) (−1,−4, 0) + t(−1, 0, 1), t ∈ R
(c) (−1, 4, 0) + t(−1, 0, 1) + s(−1, 1, 0), t, s ∈ R
(d) (−1,−4, 0) + t(−1, 0, 1) + s(−1, 1, 0), t, s ∈ R
2. Sejam ~v =
[
1 1 1
]T , ~u = [−1 −1 1]T e A tal
que A~v =
[
1 2 1 2
]T e A~u = [2 2 2 1]T .
Calcule a terceira coluna da matriz A:
(a)
[
3
2 2
3
2
3
2
]T
(b)
[− 12 0 − 12 12]T
(c)
[
1 32
3
2
]T
(d)
[
1 − 12 12
]T
3. Assinale o sistema cujo conjunto-solução é um plano:
(a)
 x+ y + z + w = 0x− y + z − w = 0
3x+ y + 3z + w = 0
(b)
 x+ 2y + z + 2w = 02x+ 4y + 2z + 4w = 0
3x+ 6y + 3z + 6w = 0
(c)
{
x+ y + z = 0
x− y + z = 0
(d)
{
x+ y + z + w = 0
4. Sejam u,v e w, vetores de R5.
(I) Se u + v − w = 2u + v + 3w, então o conjunto
{u,v} é linearmente dependente.
(II) Se 2v− u = w− v, então o conjunto {u,v,w} é
linearmente dependente.
(a) I é falsa e II é verdadeira
(b) Ambas são falsas
(c) I é verdadeira e II é falsa
(d) Ambas são verdadeiras
5. Dizemos que um polinômio de grau 2, na forma
p(x) = ax2 + bx + c, interpola dois pontos (x1, y1)
e (x2, y2), se p(xi) = yi, para i = 1, 2. O coeficiente a
é chamado de coeficiente dominante. Dados os pon-
tos ~u = (1, 2) e ~v = (3, 4), classifique as afirmativas
abaixo:
(I) Existe um polinômio de grau 2, com coeficiente
dominante positivo, que interpola ~u e ~v.
(II) Existe um polinômio de grau 2, com coeficiente
dominante negativo, que interpola ~u e ~v.
(a) Ambas são verdadeiras
(b) I é falsa e II é verdadeira
(c) Ambas são falsas
(d) I é verdadeira e II é falsa
6. Uma loja virtual é composta por n produtos e P é
uma matriz, m × n, onde Pij registra o valor que o
cliente i gastou no produto j. Seja ~1 = (1, . . . , 1),
um vetor de dimensão n; e seja ~1k = (
1
k , . . . ,
1
k ), um
vetor de dimensão k, onde k ∈ N. Assinale abaixo a
expressão que corretamente representa o gasto médio
de todos os clientes da loja:
(a) (P~1)T ~1m
(b) (P~1)~1n
(c) (P~1)T ~1n
(d) (P~1)~1m
7. Considere as matrizes A, 3× 3, e B, 4× 3.
(I) Se o sistema A ~X = ~0 tem solução única,
então o sistema A ~X = ~v sempre tem solução
qualquer que seja ~v ∈ R3.
(II) Se o sistema B ~X = ~0 tem solução única,
então o sistema B ~X = ~w sempre tem solução
qualquer que seja ~w ∈ R4.
(a) I é verdadeira e II é falsa
(b) I é falsa e II é verdadeira
(c) Ambas são verdadeiras
(d) Ambas são falsas
8. Assinale a equação que quando adicionada ao sistema
indeterminado abaixo transforma-o em um sistema de
solução única:
{
x+ y + z = 1
x− y − z = 2
(a) x+ 3y + z = 6
(b) x+ 3y + 3z = 3
(c) 3x+ y + z = 4
(d) 3x− y − z = 5
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
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Gabarito Pág. 10