exercícios - espaço vetorial
TRANSCRIPT
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V ={(x, y)R2|y0}
R
2
V
(x, y) + (x, y) = (x + x, yy)
(x, y) = (x, y)
R
2
R
V ={(x, y)R2| x, y R}
R
(x1, y1) + (x2, y2) = (x1+x2, y1+y2) (x, y) = (x,y)
(x1, y1) + (x2, y2) = (x1, y1) (x, y) = (x, y)
(x1, y1) + (x2, y2) = (2x1
2y2,
x2+y1) (x, y) = (x,y)
(x1, y1) + (x2, y2) = (x1+x2 1, y1+y21) (x, y) = (x + 1, y + 1) V ={xR| x >0}
x+y = xy (x, y) =x R
V
R
n n
a a+ba b
:a, bR
V ={(a,a,...,a)R; aR}
{(1, a , b) :a, bR}
{(x, x+ 3); xR}
{(a, 2a, 3a) :aR}
S
V
V = R3
S={(x,y,z)R3 | x = 0}
V = R3
S={(x,y,z)R3 | x
}
V =M(2, 2) S={
A
M(2, 2)|
A }
V =P3 S={pP3| p(t)0, tR}
V =P S={pP| p(0) = 2p(1)}
R4
W ={(x,y,z,t)R4 | x +y = 0 e z t= 0}
U={(x,y,z,t)R4 | 2x+y t= 0 e z= 0} M(2, 2)
V ={
a bc d
a,b,c,dR b= c}
W ={
a bc d
a,b,c,dR b= c+ 1}
W ={(x, y)R2 | (x 1)(y 1) = 1} R2
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W ={(x,y,z)R3 | x+y z = 1} W ={(x,y,z)R3 | x2 +y2 = 1} R3 v= (1, 1, 2,1)
v1 = (1, 0, 0, 0), v2 = (0, 1, 1, 0), v3= (0, 0, 1, 0), v4= (1, 0, 0, 1)
W M(2, 2)
v1 =
0 10 0
, v2 =
0 00 1
, v3 =
1 01 0
W R3
S={(1, 0, 0), (1, 0, 1)}
W =
{(x,y,z)
R3
|x+y+z = 0
}
1, 1 t, (1 t)2 (1 t)3 P3 (a, b) (c, d) adbc= 0 adbc= 0
R4
S= [(1, 1,2, 4), (1, 1,1, 2), (1, 4,4, 8)]
(2/3, 1,1, 2)
S
(0, 0, 1, 1)
S
W
M(2, 2)
W =
2a a+ 2b
0 a b| a, bR
0 20 1
W?
0 23 1
W?
W
M(3, 2)
0 01 10 0
,
0 10 11 0
0 10 00 0
0 23 45 0
W
x= (1, 0, 0) ={(1, 1, 1), (1, 1, 0), (1, 0,1)} V =M(2, 2) W
1 54 2
,
1 11 5
,
2 45 7
,
1 75 1
W
R4
v1= (1,1, 0, 0), v2 = (0, 0, 1, 1), v3 = (2, 2, 1, 1, )
v4 = (1, 0, 0, 0)
(2,3, 2, 2)[v1, v2, v3, v4]
[v1, v2, v3, v4]
[v1, v2, v3, v4] = R4
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R3
v1 = (1, 1, 0), v2 = (0,1, 1) v3 = (1, 1, 1)[v1, v2, v3] = R
3
W = [v1, v2, v3] R3
v1 = (2, 1, 0), v2 = (1, 0, 1), v3 = (1, 1, 1)
W
v= (, 2,2)
W
P2 2
p1(t) = 1 + 2t+t2, p2(t) = 2 + 4t+ 2t2
p1(t) =t+t2, p2(t) = 2, p3(t) = 1 + 2t
2
p1(t) = 1 +t, p2(t) = 2 +t, p3(t) = 2t
2
={(0, 2, 2), (0, 4, 1)} W ={(x,y,z)R3 | x = 0}
R3
S={(1, 0, 2), (0,1, 2), (2, 1, 1), (2, 2, 2)}
v1 = (1, 1, 1, 1), v2 = (1, 2, 3, 2), v3= (2, 5, 6, 4), v4= (2, 6, 8, 5)
R4
W = [v1, v2, v3, v4] R
4
v1 = (1,1, 0, 0), v2 = (0, 0, 1, 1), v3 = (2, 2, 1, 1)e v4 = (1, 0, 0, 0) v= (2,3, 2, 2) W
W
W = R4
W
R
4
W1 W2 R3
W1 ={(x,y, 0)| x, y R} W2 ={(x,y,x y)|x, y R} dim(W1W2) dim(W1+W2)
W1W2 R3
R4
W1 ={(x,y,z,t)R4 :x+y = z t= 0} e W2 ={(x,y,z,t)R4 :x y z+t= 0}
W1, W2, W1W2 W1+W2 W1+W2 = R4 M(2, 2)
W1 =
a bb a
:a, bR
e W1 =
x yx y
:x, y R
W1, W2, W1 W2 W1+W2
M(2, 2) W1+W2
V1 ={(x,y,z) R3 | x+ 2y +z = 0} V2 R3 =V1 V2
R3
V1 + V2 = R3
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W = [v1, v2, v3] P2
v1 = 1, v2 = 1 t+t2 e v3 = 1 2t+ 2t2
v1, v2 v3
W
P2 v1 v2
R3 = [(1, 0, 0)] [(1, 1, 0), (0, 1, 1), (1, 0,1)]
W1 ={(x,y,z) R3 : x+ 2y+ z = 0} W2 1
R3 =W1 W2 dimW2
{(1, 0), (0, 1)} 1 ={(1, 1), (1, 1)} 2 ={(
3, 1), (
3,1)} 3 ={(2, 0), (0, 2)}
R2
(i)[I]1 (ii)[I]1
(iii)[I]2 (iv)[I]3
v= (3,2) (i) (i)1 (iii)2 (iv)3
v
1
[v]1 =
40
v
(i) (ii)2 (iii)3
[I]
= 1 1 0
0 1 11 0 1
[v] [v] =
12
3
[v] [v]=
12
3
R3
={(1, 1, 1), (1, 1, 0), (1, 0,1)} e ={(1, 0, 0), (0, 1, 0), (0, 0, 1)}
[I]
[I] [I]
[I] =I3
v= (1, 2,1)
P2 ={1, 1 +t, t2} ={2,t, 1 +t2}
[I]
[I]
[I]
[I]
=I3
v= t2 +t 2
[v] v R3
[I]
[v] =
12
3
e [I] =
1 1 00 1 1
1 0 1