Garcia Jesus

8/18/2019 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.


9/174

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} ·


10/174

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


11/174

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, . . . ;


12/174

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


19/174

[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


24/174

⇒ 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) ⇒


25/174

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


26/174

{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 )


27/174

{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)]

Garcia Jesus

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