Triptico Mérida. Antonio Macías, Jesus Candelario y Lucia Garcia
Garcia Jesus
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V ∗
Av = λv
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G
∗
∗
(u, v)
G
u∗v
G
∗
u∗(v∗w) = (u∗v)∗w
u,v,w ∈ G
e ∈ G
u ∈ G u∗e = e∗u = u
u ∈ G
u ∈ G u∗u = u∗u = e
Z
s
−s
R
s −s
R̄ R − {0}
s ∈ R̄ 1s
A
−A.
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π
2
R
R
R
R.
G
∗
u∗v = v∗u
u, v ∈ G
R2
(x, y)
R2
A1, A2, A3
A4
A1(x, y) = (x, y)
A2(x, y) = (−x, −y)
A3(x, y) = (−x, y)
A4(x, y) = (x, −y)
2 × 2
2 × 2
I =
1 00 1
, A1 =
0 −11 0
,
A2 =
−1 00 −1
, A3 =
0 1−1 0
K
+
·
K
+
K − {0} ·
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a
b
c ∈ K a·(b + c) = a·b + a·c
Q
R
C
K
R
C
K 1
K 2
a
b
c
. . .
a1
a2
. . .
1ra
V
K
∗
·
V
∗
·
u ∈ V a ∈ K a·u ∈ V
u ∈ V 1·u = u 1
K
u ∈ V
a, b ∈ K a·(bu) = (ab)·u
u, v ∈ V a ∈ K a·(u + v) = a·u + a·v
u ∈ V a, b ∈ K (a + b)·u = a·u + b·u
2a
∗
·
∗
·
∗
u, v ∈ V u∗v = w ∈ V,
u, v ∈ V u∗v = v∗u
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u,v,w ∈ V u∗(v∗w) = (u∗v)∗w
e ∈ V
∗
u ∈ V, u∗e = e∗u = u
v ∈ V v ∈ V
v∗v = v ∗v = e
·
u ∈ V a ∈ K a·u ∈ V
u, v ∈ V a ∈ K a·(u∗v) = a·u∗a·v
·
∗
u ∈ V a b ∈ K, (a + b)·u = a·u + b·u
a
b ∈ K v ∈ V a·(bv) = (ab)·v
v ∈ V, 1·v = v
3ra
V
∗
V
V
∗ : V × V → V
·
K
V
V
· : K ×V → V
V
∗
·
V
V × V = {(u, v) | u ∈ V y v ∈ V }
K × V = {(a, v) | a ∈ K y v ∈ V }
∗ : (u, v) → u∗v · : (a, v) → a·v
K
V
V
∗
·
U , V , . . . , o U 1, U 2, . . . ;
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e ∈ V ∗
e ∈ V
e
e∗e = e
e
e∗e = e
e
V
e
e = e
e
e
∗
v ∈ V v v∗v = e
w ∈ V v ∈ V v∗w = e,
v∗
v = v∗
w
v
v∗(v∗v) = v∗(v∗w)
(v∗v)∗v = (v∗v)∗w
e∗v = e∗w
v = w,
v
v
−v
0·u = 0
K
u ∈ V
u
u∗
(0·
u) = (1·
u)∗
(0·
u)
·
(1·u)∗(0·u) = (1∗0)·u
·
1∗0
1∗0 = 1 + 0 = 1
u∗(0·u) = (1∗0)·u = u
0·u = 0
0
a·0 = 0
0
a ∈ K
a·u,
a·u∗a·0 dist.= a·(u∗0) = au,
a·0 = 0
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u
v ∈ V a b ∈ K
(−1)v = −v
a·u = 0
a = 0
u = 0
a·u = a·v
a = 0
u = v
a·u = b·u
u = 0 a = b
(
−a)·u = a·(
−u) =
−(a·u)
−(u∗v) = (−u)∗(−v)
u∗u = 2u u∗u∗u = 3u
∗
K n = {(a1, a2, . . . , an) ai ∈ K }
u, v ∈ K n u, v u = (b1 b2
. . .
bn)
v = (c1
c2
. . .
cn),
v+u = (c1, c2, . . . , cn)+(b1, b2, . . . , bn) def
= (c1+b1, c2+b2, . . . , cn+bn).
e
0
K n
0 = (0, 0, . . . , 0 n
)
u ∈ K n, u = −u u = (a1, . . . , an),−u = −(a1, . . . , an) = (−a1, . . . , −an)
a ∈ K u ∈ K n u = (a1, a2, . . . , an),a·u = a·(a1, a2, . . . , an)
def = (aa1, . . . , a an)
a·u ∈ K n
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K n
z =| z | eiϕ
z =| z | eiϕ | z |
eiϕ
ϕ
K
n
K
y+ay+by = 0
a, b ∈ C
ϕ1
ϕ2
ϕ1 + ϕ2
ϕ1 + ϕ2
s ∈ C ϕ1
sϕ1
sϕ1
(sϕ1) + a(sϕ1) + b(sϕ1) = sϕ1 + asϕ
1 + bsϕ1
= s[ϕ1 + aϕ1 + bϕ1]
= s0
= 0.
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V = {0}
K
R2
V = {(x, y) ∈ R2 | y = λx, λ = cte.}
u = (x1, y1)
v = (x2, y2) ∈ V, u + v ∈V,
u + v = (x1, y1) + (x2, y2) = (x1, λx1) + (x2, λx2)
= (x1 + x2, λx1 + λx2) = (x1 + x2, λ(x1 + x2)) ∈ V.
u = (x1, y1)
v = (x2, y2) ∈ V u+v = v+u
u + v = (x1, y1) + (x2, y2) = (x1, λx1) + (x2, λx2)
= (x1 + x2, λx1 + λx2) = (x2 + x1, λx2 + λx1)
= (x2, λx2) + (x1, λx1)
= v + u.
u = (x1, y1)
v = (x2, y2)
w = (x3, y3) ∈ V
u + (v + w) = (u + v) + w
u + (v + w) = (x1, λx1) + (x2 + x3, λ(x2 + x3))
= (x1 + (x2 + x3), λx1 + λ(x2 + x3))
= ((x1 + x2) + x3, λ(x1 + x2) + λx3)
= [x1 + x2, λ(x1 + x2)] + (x3, λx3)
= (u + v) + w.
(0, 0) ∈ R2 (0, 0) = (0, λ0)
u = (x, y) ∈ V
u + 0 = (x, λx) + ( 0, λ0) = (x + 0, λx + λ0) = (x, λx) = u.
u ∈ V u ∈ V u + u = 0
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u = (x, y) ∈ V −u = −(x, y) = −(x,λx) =(−x, −λx) V
u + (−u) = (x,λx) + (−x, −λx) = (x + (−x), λx + (−λx))= (0, 0) = e.
u = (x, y) ∈ V a ∈ R, au ∈ V,
au = a(x, y) = a(x,λx) = (ax,aλx) = (ax,λax) ∈ V.
u = (x1, y1)
v = (x2, y2) ∈ V a ∈ R a·(u + v) = a·u + a·v
a·(u + v) = a·[(x1, y1) + (x2, y2)] = a·[(x1, λx1) + (x2, λx2)]
= a·[x1 + x2, λx1 + λx2] = [a(x1 + x2), a(λx1 + λx2)]
= [ax1 + ax2,λax1 + λax2] = (ax1,λax1) + (ax2,λax2)
= a·(x1, λx1) + a·(x2, λx2)
= a·u + a·v.
u = (x1, y1) ∈ V a, b ∈ K (a + b)·u = a·u + b·u
(a + b)·u = (a + b)·(x1, y1)
= (a + b)·(x1, λx1)
= ((a + b)x1, (a + b)λx1)
= (ax1 + bx1,λax1 + λbx1)
= (ax1,λax1) + (bx1,λbx1)
= a·(x1, λx1) + b·(x1, λx1)
= a·u + b·u.
u = (x, y) ∈ V a, b ∈ R
a·(bu) = (ab)·u,
a·(bu) = a·[b(x,λx)] = a·(bx, bλx) = (abx, abλx)
= (ab)·(x,λx)
= (ab)·u.
u = (x, y) ∈ V 1 ∈ R,1·u = u,
1·u = 1·(x,λx) = (1x, 1(λx)) = (x,λx) = u.
R
2
y = λx
R2
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2a
R3
V = {(x,y,z) ∈ R3 | y = ax + by + cz = 0, a,b,c ∈ R, fijos}
u = (x1, y1, z1)
v = (x2, y2, z2) ∈ V u + v ∈ V
u + v = (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2)
V
a(x1 + x2) + b(y1 + y2) + c(z1 + z2)
= ax1 + ax2 + by1 + by2 + cz1 + cz2
= (ax1 + by1 + cz1) + (ax2 + by2 + cz2)= 0 + 0 = 0.
u = (x,y,z) ∈ V, −u = −(x,y,z) = (−x, −y, −z)
−u u
−u
u = (x1, y1, z1) ∈ V
λ ∈ R
λu ∈ V λu = λ(x,y,z) = (λx,λy,λz)
a(λx) + b(λy) + c(λz) = λ(ax + by + cz) = λ0 = 0.
R3
W,
[a, b] ⊂ R
F, G ∈ W λ ∈ R
(F + G)x = F (x) + G(x) y (λF )x = λ(F (x))
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0(x) =0
F −F (−F )x = −(F (x))
r
s
R
M rs(R)
(aij )
(bij ) ∈ M rs(R) λ ∈ R
(aij ) + (bij ) def
= (aij + bij)
λ(aij ) def
= (λaij )
0 ∈ M rs(R)
S 1 = {(a, b) ∈ R2 : a ≥ 0}
S 2 = {(a, b) ∈ R2 : a = b}
n × n
[a, b]
f
f (a) = f (b) = 0
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[a, b]
f
f (c) = 0
a < c < b
R3
{3x − y − 3 = 0 ∩ y − 3z − 3 = 0},
[a, b]
M
V
M
V
V
M
V
V
∀ u ∈ M ∀ a ∈ K au ∈ M ∀ u, v ∈ M, u∗v ∈ M
M
M
V
M
V
V
∀ u, v ∈ M a1, a2 ∈ K
u, v
M,
(a1u + a2v) ∈ M
R3
M 1 = {(a,b,c) ∈ R3 : c = 0}
M 2 = {(a,b,c) ∈ R3 : b = 0}
M 3− = {(a,b,c) ∈ R3
: a = 0} L1 = {(a,b,c) ∈ R3 : a = b = 0}
L2 = {(a,b,c) ∈ R3 : a = c = 0}
L3 = {(a,b,c) ∈ R3 : b = c = 0}
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N = {(a,b,c) ∈ R3 : a = b = 2c}
R3
V
V
M
N
V
M + N = {v ∈ V : v = v1 + v2, v1 ∈ M y v2 ∈ N }M ∩ N = {v ∈ V : v ∈ M y v ∈ N }
M + N y M ∩ N V
M + N
w, u ∈ M + N, w = w1 + w2 w1 ∈ M w2 ∈ N ; u = u1 + u2 u1 ∈ M y u2 ∈ N ; a1, a2 ∈ K
a1w + a2u = a1(w1 + w2) + a2(u1 + u2)
= a1w1 + a1w2 + a2u1 + a2u2
= (a1w1 + a2u1) + (a1w2 + a2u2) ∈ M + N
a1w1 + a2u1 ∈ M a1w2 + a2u2 ∈ N
M
N
V.
M ∩ N
u, v ∈ M ∩ N, u, v ∈ M y u, v ∈ N, a1, a2 ∈ K, a1u + a2v ∈ M ∩ N,
(a1u + a2v) ∈ M (a1u + a2v) ∈ N.
S 1 = {(x, y) ∈ R2 : x = 2y}
S 2 = {(x,y,z) ∈ R3 : x + y + z = 0}
S 3 = {(x,y,z) ∈ R3 : x = y, 2y = z}
S 4 = {(aij ) ∈ M nn : aij = a ji}
S 5 = {(aij ) ∈ M nn : aij = 0 cuando i = j}
S 6 = (aij ) ∈ M 22 : (aij ) =
a b
−b a
S 7 =
(aij ) ∈ M 22 : (aij ) = x 0
0 x + 2
S 8 = {(aij ) ∈ M 33 : aij = 0 cuando i = j}
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S 9 = {P ∈ P 3 : P = 3}
S 10 = {P ∈ P n : P n(0) = 0}
S 11 = {f ∈ C [a, b] : f (a) = f (b) = 0}
S 12 = {f ∈ C [a, b] : f (c) = 0, a < c < b}
S 13 = {f ∈ C [a, b] :
b
a f (x)dx = 0}
S 14 =
{f
∈C [a, b] : ba f (x)dx =
√ 2
}
L1
L2
L3
M 1
M 2
M 3
N
N 1
N 2
M 22
N 1 = {(aij ) ∈ M 22 : a12 = 0}N 2 =
(aij) ∈ M 22 : (aij ) =
a b−b a
N 1
N 2
M 22
N 1 + N 2
M 22
N 1 ∩ N 2 M 22
R3 = M 1 + M 2
w ∈ R3
M 1
M 2
(a,b,c) = (0, b, 0) + (a, 0, c) =
1
2a,b, 0
+
1
2a, 0, c
= (a + d,b, 0) + (−d, 0, c). con d ∈ R arbitrario.
M
N
V,
0 ∈ M 0 ∈ N,
0 ∈ M ∩ N.
M ∩ N M ∩ N = {0}
w ∈ M + N
w = u1 + u2
u1 ∈ M
u2 ∈ N
w = v1 + v2
v1 ∈ M
v2 ∈ N,
u1 + u2 = v1 + v2
u1 − v1 = v2 − u2,
M
N
M
N
M
N
u1 − v1 = 0 ⇒ u1 = v1 y u2 − v2 = 0 ⇒ u2 = v2
M ∩ N = {0} w ∈ M + N
M
N
M + N
M
N
M + N = M ⊕ N
M + N
w
∈M
∩N
w = w + 0
w ∈ M 0 ∈ N w = 0 + w 0 ∈ M w ∈ N
w
w = 0
M ∩ N = {0},
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M + N = M ⊕ N ⇐⇒ M ∩ N = {0}
M 1
M 2
. . .
M n
V
M = M 1 + M 2 + · · · + M n
V,
w ∈ M, w = w1 + w2 + · · ·+ wn, wi ∈ M i.
M 1
∩M 2
∩ · · · ∩M n,
w
∈ V
w ∈ M i
i
M 1 + M 2 + · · · + M n = M 1 ⊕ M 2 ⊕ · · · ⊕ M n w ∈ M 1 + M 2 + · · · + M n
w = w1 + · · · + wn con wi ∈ M iw = v1 + · · · + vn con vi ∈ M i
⇒ vi = wi
M 1 + M 2 + · · · + M n = ⊕ni=1 M i ⇒ M i ∩ M j = {0},
(x,y,z) ∈ M 1 + L1 + N M 1 ∩L1 ∩N = {0}, M 1, L1yN
(x,y,z) = (x − 2z + 2γ, y − 2z + 2γ, 0) + (0, 0, γ ) + (2z − 2γ, 2z − 2γ, z − γ )
γ
(x,y,z)
V
v ∈ V
av
a ∈ K
Lv
Lv
V
{u1, u2, . . . , un} ={ui}ni=1 ⊂ V
Lu1+Lu2+· · ·+Lun =Lu1 ⊕ Lu2 ⊕ · · · ⊕ Lun .
Lu1 + Lu2 + · · · + Lun = Lu1 ⊕ Lu2 ⊕ · · · ⊕ Lun ,
w ∈ Lu1 + Lu2 + · · · + Lun ,
w
{u1, . . . , un} w = w1 + w2 + · · · + wn
wi ∈ Lui
{u1 u2 . . .
un} ∀ w ∈ Lu1 + Lu2 + · · · + Lun ,
w = a1u1 + a2u2 +
· · ·+ anun
w = b1u1 + b2u2 + · · · + bnun ⇒ ai = bi
0 = w − w = (a1 − b1)u1 + (a2 − b2)u2 + · · · + (an − bn)un
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{u1 u2 . . . un}
{u1 u2 . . . un} 0 = a1u1 + a2u2 + · · · + anun ⇒ ai = 0 i = 1, . . . , n .
{u1 u2 . . .
un} {u1 u2 . . . un}
(5, 0, 0), (0, −1, 0) (0, 0, 12 )
(0, 0, 0) = a1(5, 0, 0) + a2(0, −1, 0) + a3(0, 0, 12
)
= (5a1, 0, 0) + (0, −a2, 0) + (0, 0, a32
)
= (5a1, −a2, a32
)
⇒
5a1 = 0 ⇒ a1 = 0−a2 = 0 ⇒ a2 = 0
a32 = 0 ⇒ a3 = 0
{x, y, z} V { x + y
y − z 2x − z }
0 = a1(x + y) + a2(y − z) + a3(2x − z)= a
1x + a
1y + a
2y
−a
2z + 2a
3x
−a
3z
= (a1 + 2a3)x + (a1 + a2)y − (a2 + a3)z
{x, y, z}
⇒
a1+ 2a3 = 0a1+ a2 = 0a2+ a3 = 0
⇒ a1 = a2 = a3 = 0
{x, y} { x + y, −y, −x }
0 = a1
(x + y)−
a2
y−
a3
x = a1
x + a1
y−
a2
y−
a3
x
= (a1 − a3)x + (a1 − a2)y
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⇒ a1 − a3 = 0a1 − a2 = 0
⇒ a3 = a1
a2 = a1
0 = a1(x + y) − a1y − a1x para toda ai ∈ K
{ x + y, −y, −x }
{ 1, x, x2 } V
{ 1 + x, x2 −1, 1 + x + x2 },
{u1, u2, . . . , uk} ∈ V uk+1 ∈ V
uk+1 ∈ Lu1 + Lu2 + · · · + Luk ⇒ {u1, u2, . . . , uk, uk+1}
{u1
u2
. . .
uk+1}
0 = a1u1 + a2u2 + · · · + akuk + buk+1 b
0 = a1u1 + a2u2 + · · · + akuk ⇒ ai = 0, i = 1, . . . , k
{ui}ki=1
{u1 u2
. . .
uk+1} b = 0
uk+1 = −1b (a1u1 + · · · + akuk) ⇒
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uk+1 ∈ Lu1 + Lu2 + · · · + Luk
{u1 u2 . . . uk+1}
{u1, u2, . . . , uk
} ⊂V
uk+1
∈V
uk+1
∈Lu1 + Lu2 +
· · ·+
Luk ⇒ {u1, u2, . . . , uk, uk+1}
V
V
V
{u1, u2, . . . , ur}
V = Lu1 + Lu2 + · · · + Lur
{u1, u2, . . . , ur}
V
V
r,
V = Lu1 + Lu2 + · · · + Lur = Lu1 ⊕ Lu2 ⊕ · · · ⊕ Lur
{u1, u2, . . . , ur} V
{v1, v2, . . . , vn}
V
{v1, v2, . . . , vn} ⊂ V. {v1, v2, . . . , vn} generan V. {v1, v2, . . . , vn}
{v1, v2, . . . , vn} ⊂ V
V = Lv1 + Lv2 + · · · + Lvn
{v1, v2, . . . , vn} {v1, v2, . . . , vn} V. V
{vi}ni=1
V
{vi}1i=1
v1,
v1 = 0
n > 1
v1 = 0 {vi}ni=1
vk
vk
1ra
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{vi}1i=1
0 = av1
v1 = 0
a = 0 v1 = 0 av1 = 0
a ∈ k {vi}1i=1
2da
{vi
}ni=1
0 = a1v1 + a2v2 + · · · + akvk + ak+1vk+1 + · · · + anvn
ak
vk
vk = − 1ak (a1v1 + a2v2 + · · · + ak−1vk−1)
vk
vk
vk = b1v1 + b2v2 + · · · + bk−1vk−1
0 = b1v1 + b2v2 + · · · + bk−1vk−1 − vk
{v1, v2, . . . , vk}
{v1, v2, . . . , vn}
V
{v1, v2, . . . , vn} {w1, w2, . . . , wr} V
r
n
vi = r
j=1 c ji w j
wk = n
s=1 dskvs,
w1 = d11v1 +· · ·+dn1 vn
{w1, v1, . . . , vn} vk k = 1, . . . , n
{w1, v1, . . . , vk−1, vk+1 . . . , vn} V ; w2
{w2
w1
v1
. . .
vk−1
vk+1
. . .
vn} V
vt
V
{w2 w1
v1
. . .
vt−1
vt+1
. . .
vk−1
vk+1
. . .
vn},
n
vs
{w1, w2, . . . , wn} V
r > n
wn+1, wn+2, . . . , wr
{w1, w2, . . . , wn}
ws
V
r = n.
M
N
V
D(M + N ) = D(M ) + D(N ) − D(M ∩ N )
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{w1 w2 . . . wr} M ∩ N M ∩ N
M
M ∩ N, v ∈ M v ∈ Lw1 + · · · + Lwr
M
{w1 w2 . . . wr
v1
v2
. . .
vs} M ∩ N
M i ∈ N N {w1 w2 . . . wr
u1
u2
. . .
um}
M
N
M + N
{w1, w2, . . . , wr, v1, v2, . . . , vs, u1, u2, . . . , um}
M + N
0 = a1w1 + · · · + arwr + b1v1 + · · · + bsvs + c1u1 + · · · + cmum
x ∈ Lw1 + · · · + Lwr + Lu1 + · · · + Lum −x ∈ Lv1 + Lv2 + · · · + Lvs,
x ∈ M ∩ N, x ∈ Lw1 + Lw2 + · · · + Lwk ,
{w1, w2, . . . , w
k}
x
{Lw1
Lw2
. . .
Lwk}
x = 0,
ai = b j = ck = 0
D(M + N ) = D(M ) + D(N ) − D(M ∩ N ) = k + m + k + n − k
M + N = M ⊕ N ⇐⇒ D(M + N ) = D(M ) + D(N )
(a, b) ∈ R2 { (1, 2), (2, 4), (1, −2) }
(x, y, z) ∈ R3 { (1, 1, 1), (0, 2, 2), (0, 0, 3) }
(x, y, z) ∈ R3 { (1, 1, 1), (0, 2, 2), (0, 0, 3), (1, 2, 3) }
1 23 4
∈ M 22
2 00 0 , 2 20 0 , 2 22 0 , 2 22 2
1 − 2x + x2 ∈ P 2 {1, 1 + x, 2 + x2 }
(a1, a2, . . . , an) ∈ Rn
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{e1 = (1, 0, . . . , 0 n
), e2 = (0, 1, 0, . . . , 0 n
), . . . , en = (0, . . . , 0, 1 n
) }
(2, 3i, −5i) ∈ C 3
{e1 = (1, 0, −i), e2 = (1 + i, 1 − i, 1), e3 = (i, i, i)} ⊂ C 3
f (t)
{ et, e−t } f (0) =1
f (0) = 2. f (t)
{ sen x, cos x, ex }
f (0) = 0, f (0) = 1
f (0) = 1.
(3+i, i)
(1+i, 2i)
(i, 1−i)
T 1 = { (a, a), (b, b) } ⊂ R2.
T 2 = { (a, 0, 0), (0, a, 0), (0, 0, a) } ⊂ R3.
T 3 =
a 00 0
,
a a0 0
,
a aa 0
,
a aa a
⊂ M 22
T 4 =
a 00 0
,
a a0 0
,
a aa 0
⊂ M 22
T 5 = { 1, 2x, 3x2 } ⊂ P 2.
T 6 = { 1, x, x + 1 } ⊂ P 1.
T 7 = { 1, x, x + 1 } ⊂ P 2.
{ (a, b), (c, d), (e, f ) } ⊂ R2.
{ (a, b), (−a, b + 1) } ⊂ R2.
{(a,b,c), (a,
−b,
−c), (
−1, 1, 1)
} ⊂R3.
{ (1, 1, 1), (0, 2, 2), (0, 0, 3) } ⊂ R3.
1 11 1
,
1 01 1
,
1 00 1
⊂ M 22
1 21 3
,
−2 32 −5
,
2 42 6
,
1 11 1
⊂ M 22
1 00 0
,
1 10 0
,
1 11 0
,
1 11 1
⊂ M 22
0 12−3 0
,
1
2 00 −3
⊂ M 22
{ 5, 1 + x, −x2 } ⊂ P 2.
{ 1 + x, 1 + x2, 3x2 } ⊂ P 2.
{ x, 2x2
, 3x
3
} ⊂ P 3. { 2, 2 + x, 3 − x2 + 2x, x3 − x2 + x + 1 } ⊂ P 3.
{ sen x, cos x } ⊂ [a, b]
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{ sen x, sen 2x } ⊂
[a, b]
{ 2x, 3√ x, 4 3√ x } ⊂ [a, b]
{ ex, e2x } ⊂ [a, b]
f 1
f 2
f +
f = 0
f 1(0) = 1, f 1(0) = 0, f 2(0) = 0, f 2(0) = 1
f 1
f 2
sen x
cos x
{u1, u2, . . . , un}
{u2, . . . , un}
{ (a1, a2, a3), (b1, b2, b3), (c1, c2, c3) } ⊂ R3
α
{ (1, 1, 1), (1, α, 0), (α, 1, 1) } ⊂ R3
P 3
P 3.
P 3
{v1, v2, . . . , vn} r < n
{v1, v2, . . . , vr}
{ v1, v2, v3 } { v1 + v2 v1 + v3, v2 + v3 }
{ 1−x2, 1+ x }
C 2
C
(1 + i, 2i)
(i, 1 − i)
(1, i)
(i, −1)
K n = {(a1, a2, . . . an) ∈ K },
z =|z | eiϕ
K
n,
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y+ ay+ by = 0
a, b ∈ C,
V
V
V
f
f (0), f (1), f (2), f (3)
V
f
f (−2), f (−1), f (1), f (2)
V = K 3
K
W
(1, 0, 0)
U
(1, 1, 0)
(0, 1, 1) .
V
W
U.
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A
V, W
v1, v2 ∈ V A(v1 + v2) = A(v1) + A(v2)
v ∈ V a ∈ K A(av) = aA(v)
V,
A;
V
W
v ∈ V V
V
W
A1, A2
V
W,
v ∈ V a ∈ K
(A1 + A2)v def = A1v + A2v
(aA1)v = a[A1(v)]