Teste 2008.2 Matéria da P1

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NOME:
1 (b) (e)
2 (d)
3 (c)
4 (b) (c)
5 (b) (c) (e)
6 (b)
7 (e)
8 (a) (d)
9 (b)
Questa˜o 1. Seja W o subespac¸o soluc¸a˜o de um sis-
tema linear homogeˆneo. Adicionando uma equac¸a˜o,
dim (W) (Marque *todas* as alternativas corretas)
(a) Pode aumentar
(b) Pode diminuir
(c) Certamente aumenta
(d) Permanece a mesma
(e) Pode permanecer na mesma
Questo˜es 2-3. Considere os vetores
(x, y, z, w) ∈ R4 que sa˜o soluc¸a˜o de{
x+ z − w = 0
−z + w = 0
O conjunto soluc¸a˜o e´ gerado por
(a) {(1,0,-1,1), (0,1,0,0), (1,0,1,0)}
(b) {(0,1,0,0), (0,0,1,1), (1,0,-1,1)}
(c) {(1,0,-1,1), (0,1,0,0) , (1,0,0,0)}
(d) {(0,1,0,0), (0,0,1,1)}
(e) {(1, 0, -1, 1), (0,1,0,0)}
questa˜o 2
e a dimensa˜o deste subespac¸o e´
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
questa˜o 3
Questa˜o 4. As coordenadas [q]β do vetor q(x) =
x2 + x− 1 na base β = {1, 1− x, x2 − 1} sa˜o
(a) [1,−1, 0]T
(b) [1,−1, 1]T
(c) [1,−1, 1]T
(d) [1, 1, 1]T
(e) [0, 1, 1]T
Questa˜o 5. Marque todos os conjuntos que sa˜o
subespac¸os
(a) {f(x) = a sen(x)− 1, a, ∈ R} ⊂ F (R;R)
(b) {A | aij = 0,∀i > j} ⊂ Mn×n
(c) {(x, y, z) |x+ z = y} ⊂ R3
(d) {x |Ax = b} ⊂ Rn, b 6= 0
(e) {x |Ax = 0} ⊂ Rn.
Questa˜o 6. Considere o subespac¸o de R4
{r(1, 0, 2, 1) + s(2,−1, 1, 0) + t(−4, 3, k, 2), s, r, t ∈ R}
O valor de k para que o subespac¸o tenha dimensa˜o
2 e´
(a) k = 0 (b) k = 1 (c) k = −4 (d) k = 5
(e) k = −3
Questa˜o 7. Quais dentre os seguintes conjuntos de
vetores sa˜o bases de R3 (Marque *todas* as al´ıneas
corretas)
(a) {(1, 1, 1), (1, 2, 3)}
(b) {(1, 1, 1), (0, 0, 0), (3, 0, 1)}
(c) {(1, 1, 1), (1, 2, 3), (3, 0, 1), (1, 0, 2)}
(d) {(1, 1, 1), (2, 2, 2), (3, 0, 1)}
(e) {(1, 1, 1), (0, 1, 2), (0, 0, 7)}
Questa˜o 8. Considere o conjunto
H =
{
(x, y, z, w) ∈ R4 |x+ 2y − 3w = 0 e y − z + w = 0}
e assinale *todas* as afirmac¸o˜es corretas
(a) H = 〈(−2, 1, 1, 0), (5,−1, 0, 1)〉
(b) H e´ um subespac¸o de R4 de dimensa˜o 1.
(c) H na˜o e´ um subespac¸o de R4
(d) H =(x, y, z, w) ∈ R4
∣∣∣∣∣∣∣∣
[
1 2 0 −3
0 1 −1 1
]
x
y
z
w
 = 0

Questa˜o 9.
Seja β = {(0, 1, 2), (0, 2, 1), (1, 1, 1)} uma base de
R3, [u]β = [1, 2, 3]T as coordenadas do vetor u na
base β e v = (−4, 2, 1) um vetor de R3. A terceira
coordenada de [u+ v]β e´
(a) 0 (b) −1 (c) −3 (d) 7 (e) −4