4a Lista de Algebra Linear - gabarito

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4a Lista de A´lgebra Linear
1) a) [T ]BB′ =
[
1 1 0
0 0 1
]
b) [T ]BB′ =
 1 11 0
1 −1

c) [T ]BB′ =
[
2 3
]
d) [T ]BB′ =
 12
3

2) [T ]BB′ =
[
4
11
1
11
7
11
9
11
5
11
−9
11
]
3) a) [T ]B =

1 0 1 0
0 1 0 1
2 0 1 0
0 2 0 1

b) Tr[T ]B = 4
4)
[T ]βγ =
 0 1 −20 0 2
0 0 0
 [T ]γβ =
 0 1 10 0 2
0 0 0

5) a) T (x, y) = (−6x+ 5y,−5x+ 5y)
b) T (a+ bt+ ct2) = (2a+ b) + (a− 3c)t+ (−a+ b+ 2c)t2
c) T
(
a b
c d
)
= (a− d,−2b+ 2c+ d,−a+ b+ 2c)
6) a) T (a+ bt+ ct2) = (a, b, c)
b) [T ]BB′ =
 1 0 −10 1 −1
0 0 1

c) T−1(a, b, c) = (a+ bt+ ct2)
7) a) Autovalores: 2 e 3
Autovetores: {x(1, 1
2
) ∈ R2/x ∈ R} e {x(1, 1) ∈ R2/x ∈ R}
1
b) Autovalores: 1 e 4
Autovetores: {x(1,−1
2
) ∈ R2/x ∈ R} e {x(1, 1) ∈ R2/x ∈ R}
c) Autovalores: 4
Autovetores: {x(1, 1) ∈ R2/x ∈ R}
d) Na˜o possui.
e) Autovalores: 1 e 4
Autovetores: {x(1, 0, 0)+y(0, 1,−1) ∈ R3/x, y ∈ R} e {x(1, 1, 2) ∈
R3/x ∈ R}
f) Autovalores: -1, 1 e 2
Autovetores: {y(0, 1,−1
3
) ∈ R3/y ∈ R}, {x(1,−1,−1) ∈ R3/x ∈
R} e {z(0, 0, 1) ∈ R3/z ∈ R}
g) Autovalores: 1
Autovetores: {x(1, 0, 0) + z(0, 0, 1) ∈ R3/x, z ∈ R}
8) T (4, 1) = (8, 11)
9) a) T (x, y) = (x, 2x+ 3y)
b) T (x, y) = (−2x+ 5
2
y, 3y)
10) Demonstrac¸a˜o:
T (u) = λu
T (v) = λv
T (αu − βv) = T (αu) − T (βv) = αT (u) − βT (v) = α(λu) − β(λv) =
λ(αu− βv)
11) a) T (0, 3) = (2, 10)
b) T (x, y) = (5
3
x+ 2
3
y,−2
3
x+ 10
3
y)
c) [T ] =
[
2 0
0 3
]
12) Respostas na lista.
13)
14) Na˜o.
15)
16) < u, v >= 3
||u|| = √2
2
||v|| = 3
d(u, v) =
√
5
cos(θ) =
√
2
2
Normalizado de u+ v = ( 3√
17
, −2√
17
, 2√
17
)
17) < A,B >= 3
||A|| = √40
||B|| = √15
Normalizada de A =
[
1√
40
3√
40
−1√
40
2√
40
0 5√
40
]
Normalizada de B =
[
0 2√
15
1√
15−1√
15
3√
15
0
]
18) < A,B >= 1
||A|| = √3
||B|| = 1
d(A,B) =
√
2
cos(θ) =
√
3
3
19) < f, g >= 1
< f, h >= 11
12
||g|| = 1
||h|| =
√
1
5
Normalizado de g = 3t− 2
Normalizado de h =
√
5t2
d(f, g) =
√
28
3
d(f, h) =
√
47
10
20) < f, g >= −1
< f, h >= 0
||g|| = √13
||h|| = 1
3
Normalizado de g = 3√
13
t− 2√
13
Normalizado de h = t2
d(f, g) =
√
20
d(f, h) =
√
6
21)
22)
23) w = (2
3
,−2
3
, 1
3
)
24)
25) q = 2
3
− 2
3
t+ 1
3
t2
26) a) m = 2
b) m pode assumir qualquer valor real.
27) Mostrar que < u− kv, v >= 0
28) B′ = {( 1√
5
, 2√
5
), ( 2√
5
,− 1√
5
)}
29) B′ = {( 1√
2
, 1√
2
, 0), ( 1√
6
,− 1√
6
, 2√
6
), (−
√
3
3
,
√
3
3
,
√
3
3
)}
4