livro-ALGLIN.pdf

7/28/2019 livro-ALGLIN.pdf

1/265

u0

v u + vu0

v u + v

u0

v u + v

u0

v u + v

u0

v u + vu0

v u + v

u0

v u + v

u

0 v u + v


2/265


3/265


4/265


5/265


6/265


7/265


8/265


9/265


10/265

Rn

Rn


11/265


12/265


13/265

Rn

Rn

R

R2


14/265

R2

R3

R2 R3


15/265

Rn


16/265


17/265

Rn

Rn


18/265

Rn

R2

R30

Rn

Rn n n

Rn

R2

R3

Rn

R

C

R

v Rn v = (a1, . . . , an) ai R ai v

R2 (6, 8), (1, 2)

R4 (1, 2, 3, 4) (2, 7/4, 1, 2/3) R5 (1, 2, 4, 6, 8) (1, 2, 7/4, 1/3, 3)


19/265

(1, 2) (2, 1) (1, 2, 3) (2, 3, 1) (3, 1, 2)

Rn n > 3

(x,y,z) R3 x2 + y2 + z2 = 1

R4

(x, y, z, w) R4

x

2

+ y

2

+ z

2

+ w

2

= 1

R4

Rn

Rn

u = (u1, u2, . . . , un) v = (v1, v2, . . . , vn) Rn

u v u + v

u + v = (u1 + v1, u2 + v2, . . . , un + vn).

R4 (1, 1, 1/4, 2/3)+(2, 2, 3/4, 5/3) = (12, 1 +2, 1/4 + 3/4, 2/3 + 5/3) = (1, 1, 1, 1)

+ u + v (u1 + v1, . . . , un+ vn)

0

0 = (0, . . . , 0)

v + 0 = 0 + v = v v Rn


20/265

Rn

u = (u1, u2, . . . , un) k k u ku

ku = (ku1, ku2, . . . , k un).

k k

u = (1, 3, 1, 2, 3/2) 2u = 2(1, 3, 1, 2, 3/2)= (2, 6, 2, 4, 3)

w = (4, 6, 1, 3) 1/2w = 1/2(4, 6, 1, 3) = (2, 3, 1/2, 3/2)

Rn

n 3

R

R2

R3

(3, 2) R2

(1, 3, 2)R3

(3, 2)

3

2

3

2 (1, 3, 2)

1

u

v

u

v

u

v

(3, 2) R2

x y


21/265

3

2(3, 2)

1

2

3

1/2 1

1

1 v = (3, 2) 1, 5v, 0, 5v v

v 32v,12v v

v

{kv| k R} kv

v R2

R

3

v Rn

u + kv

v w

k v = kw

(2, 4, 6, 1) (1, 2, 3, 1/2) (2, 4, 6, 1) =2(1, 2, 3, 1/2) (1, 2, 3, 1/2) = 1/2(2, 4, 6, 1)

{k(1, 1, 1)| k R} {m(4, 4, 4)| m R}


22/265

(k, k, k) k/4(4, 4, 4) m = k/4

0

0 = 0w w

u

vw

z

uv

w

z

u + v + w + z

4

R4

R3

R

4

v v1, v2, . . . , vp v

v = 1v1 + 2v2 + + pvp =pi=1

ivi,

i

v = (2, 2) u = (1, 1) v = 2u u v


23/265

u = (1, 0, 0) v = (0, 1, 0) R3 z = 0 (a,b, 0) = a(1, 0, 0)+b(0, 1, 0) w = (3, 2, 0) u v w u v

(3, 3) = 3(1, 1) + 0(2, 2) = 1(1, 1) 2(2, 2)

(3, 4) (1, 1) (2, 2) (3, 4) =(1, 1)+(2, 2) , R

+ 2 = 3 + 2 = 4

+ 2 3 4

u = (2, 3, 4) v = (1, 0, 0) w = (1, 0, 1)

, R (2, 3, 4) = (1, 0, 0) + (1, 0, 1)

+ = 20 = 3 = 4

.

0 = 3 u

v

w

u = (1, 3, 4) v = (1, 1, 0) w = (1, 0, 1) , R (1, 3, 4) = (1, 1, 0) + (1, 0, 1) + = 1 = 3

= 4.

= 3 = 4 u = 3v + 4w

{v1, v

2, . . . , vp

}

v1, v

2, . . . , vp

span {v1, v2, . . . , vp} v1, v2, . . . , vp

v1, v2, . . . , vp = span{v1, v2, . . . , vp} =

pi=1

ivi

i R, i = 1, 2, . . . , p

.

{v1, v2, . . . , vp} W W = v1, v2, . . . , vp


24/265

(1, 0) (0, 1) R2 (a, b) R2 (a, b) = a(1, 0) + b(0, 1) (1, 0), (0, 1) = R2

(1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) {(1, 1, 1), (1, 1, 1)}

(1, 1, 1), (1, 1, 1) = (1, 1, 1) = (1, 1, 1)

Rn

Rn

Rn

v S S S S {v} S v v S S S

S = {(1, 2, 1), (1, 0, 1), (1, 1, 1)} (1, 2, 1) S (1, 2, 1) = 3(1, 0, 1) 2(1, 1, 1)

(1, 2, 1), (1, 0, 1), (1, 1, 1) = (1, 0, 1), (1, 1, 1) (1, 1, 1) (1, 1, 1) = 1/2(1, 2, 1) +3/2(1, 0, 1) (1, 2, 1), (1, 0, 1), (1, 1, 1) = (1, 2, 1), (1, 0, 1)

u, v, w, x, y, z

{u, v, w} {w, z} {v, y, z} {v, z}

{u, y} u = 2y {v, x, y} x + v = 2y {v, w, z} w + 2z = v {u, v, y} u = 2y {u, v, w, x} v + u + x = 0

u

v w

x

y

z


25/265

H H

w

v1, v2, . . . , vp

H = w + v1, v2, . . . , vp .

H = w + v1, v2, . . . , vp p {v1, v2, . . . , vp} p p p

r w

u

w + tu t

R

r = {w + tu; t R} t r

r

2u

u

2u

0

w + 0u = w

w + 1u

w + 2u

w 1u

w 2u

u

r = {w + tu; t R}

w + u

w = 0

u

u = 0 w + u =w + {0} = w

w = 0

r (1, 2) + t(4, 6) t = 0 (1, 2) r t = 1 (1, 2) +


26/265

1(4, 6) = (5, 8) r t = 0, 5 (1, 2) + 0, 5(4, 6) = (3, 5) r t = 1 (1, 2) 1(4, 6) = (3, 4) r t = 0, 5 (1, 2) 0, 5(4, 6) = (1, 1) r

R4

(2, 3, 4, 5) (1, 1, 1, 1) (2, 3, 4, 5) + t(1, 1, 1, 1) (1, 2, 1, 2) (3, 4, 3, 4) u = (3, 4, 3, 4) (1, 2, 1, 2) = (2, 2, 2, 2)

(1, 2, 1, 2) + t(2, 2, 2, 2) u = (1, 2, 1, 2) (3, 4, 3, 4) =(2, 2, 2, 2) (1, 2, 1, 2) + t(2, 2, 2, 2) w = (1, 2, 1, 2) (3, 4, 3, 4) (3, 4, 3, 4)+t(2, 2, 2, 2) (3, 4, 3, 4) + t(2, 2, 2, 2) (1, 1, 1, 2) (1, 0, 1, 0) + (2, 1, 2, 1) t R (1, 1, 1, 2) = (1, 0, 1, 0)+t(2, 1, 2, 1)

1 + 2t = 1

t = 11 + 2t = 1t = 2

.

t = 1 t = 2

(1, 2, 1)+(2, 6, 4) (0, 5, 1)+(1, 3, 2)

s t (1, 2, 1) + s(2, 6, 4) = (0, 5, 1) +t(1, 3, 2)

t = 1 + 2s3t = 3 6s2t = 2 + 4s .

t = 1 2s

ax + by + c = 0

R2 2x

3y = 6

y = t

x = 3 + 3/2y = 3 + 3/2t

(x, y) = (3, 0) + t(3/2, 1)

R2 y = 7 x = t y = 7 (x, y) = (0, 7) + t(1, 0)

y = t t = 7 x


27/265

R3

R3

R3

2z y = 1x + y + z = 0 z = t y = 2t 1 z = t y = 2t 1 x + (2t 1) + t = 0 x = 3t + 1 (x, y, z ) =t(3, 2, 1) + (1, 1, 0) y = t z = t/2 + 1/2 x = 3/2t 1/2 (x,y,z) = t(3/2, 1, 1/2) + (1/2, 0, 1/2)

x = s y z z = (1 s)/3 y = (2s 1)/3 (x, y, z ) = ( 1, 2/3, 1/3)s + (0, 1/3, 1/3) (3, 2, 1) (3/2, 2, 1/2) (1, 2/3, 1/3) (0, 1/3, 1/3) t = 1/3

R3

z = 1y + z = 0

z = 1

y + 1 = 0

y = 1 x x = t y = 1 z = 1 (x,y,z) = t(1, 0, 0) + (0, 1, 1)

z = t t x

w + tv

r = {w +tu; t R} u v = 3u v = 6u r = {w + sv; s R} z r z w

u r = {z + tu; t R}

w r


28/265

u v

w

w+tu+sv s, t R u, v = {w+tu+sv; s, t R

} t s

0

vsv

u

tutu + sv

w

w + tu + sv

= {w + tu + sv; s, t R}

w+

u, v

w = 0

{u, v}

u, v w + u, v

(1, 1, 2, 0) + t(1, 2, 1, 1) + s(1, 1, 1, 1) t = s = 0 (1, 1, 2, 0) t = 0 s = 1 (1, 1, 2, 0) + (1, 1, 1, 1) = (2, 2, 3, 1) t = 1 s = 0 (1, 1, 2, 0) +(

1, 2,

1, 1) = (0, 3, 1, 1) t = 1 s =

1 (1, 1, 2, 0 ) + (

1, 2,

1, 1)

(1, 1, 1, 1) = (1, 2, 0, 0)

u = (1, 2, 1, 1, 1), v = (2, 2, 0, 1, 1). (1, 2, 3, 4, 5) + u, v R5 {u, v}

u

v

(1, 1, 1, 1) (2, 2, 2, 2) + (1, 0, 1, 0), (0, 1, 0, 1)


29/265

s, t R (1, 3, 1, 3) = (2, 2, 2, 2) + s(1, 0, 1, 0) + t(0, 1, 0, 1)

2 + s = 12 + t = 32 + s = 12 + t = 3

.

s = 1 t = 1 {(0, 0, 1), (0, 1, 0)} y z x = 0 (0, a , b) (0, a , b) =a(0, 1, 0) + b(0, 0, 1)

(0, 1, 1) (0, 1, 0) y z (0, a , b) (0, a , b) =a(0, 1, 1) + (b a)(0, 0, 1)

R4

(1, 2, 3, 4)

(2, 3, 5, 7)

(0, 1, 0, 1) (1, 2, 3, 4) + t(2, 3, 5, 7) + s(0, 1, 0, 1) (2, 2, 2, 2) (3, 3, 3, 3) (4, 0, 4, 0) w = (2, 2, 2, 2) v = (3, 3, 3, 3) w = (1, 1, 1, 1) w = (4, 0, 4, 0) w =

(1, 2, 1, 2) w + tu + sv (1, 1, 1, 1) (2, 3, 4, 5) (2, 3, 4, 5) w = (1, 1, 1, 1) v = (2, 3, 4, 5) w = (1, 4, 3, 6) w = (2, 3, 4, 5)

w + tu + sv

R3 ax + by + cz = d

R3 2x 3y + 10z = 16 y = s z = t x = 8 + 3/2y 5z = 8 + 3/2s 5t (x,y,z) = (8, 0, 0) + s(3/2, 1, 0) + t(5, 0, 1) (8, 0, 0) + (3/2, 1, 0), (5, 0, 1)

R3 3y + 2z = 6

x x = s y = t z = 3 3/2y = 3 3/2t (x,y,z) = (0, 0, 3) + s(1, 0, 0) + t(0, 1,

3/2)

(0, 0, 3) + (1, 0, 0), (0, 1, 3/2)


30/265

{(1/2, 2, 1), (1, 4, 2)} (1/2, 2, 1) (1, 4, 2) 1 (1/2, 2, 1), (1, 4, 2) = (1/2, 2, 1) =(1, 4, 2) .

3 p

u1, u2, . . . , up w

w +t1u1+t2u2 + +tpvp ti R i = 1, . . . , p p u1, u2, . . . , up p p

w +u1, u2, . . . , up w = 0 {u1, u2, . . . , up} p

(1, 0, 0) (0, 1, 0) (0, 0, 1) 3

R3 (a,b,c) R3 (a,b,c) = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1)

(1, 1, 1) (1, 0, 1) (0, 1, 0) 2 (1, 1, 1) = (1, 0, 1)+(0, 1, 0) (1, 1, 1), (1, 0, 1), (0, 1, 0) = (1, 0, 1), (0, 1, 0)

(0, 1, 0, 0) (0, 0, 1, 0) (0, 0, 0, 1) 3 3 R4 x

u, v, w R3 3 R3

w = u + v u, v, w

u, v

(2, 3, 5, 7) +(1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3), (4, 4, 4, 4)

R4 4

(2, 2, 2, 2) = 2(1, 1, 1, 1) (3, 3, 3, 3) = 3(1, 1, 1, 1) (4, 4, 4, 4) = 4(1, 1, 1, 1)

(1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3), (4, 4, 4, 4) = (1, 1, 1, 1)

(1, 2, 3, 4, 5) + (0, 0, 0, 0, 0), (0, 0, 0, 0, 0)

S = w + u1, u2, . . . , up p

p = 0 S


31/265

p = 1 S p = 2 S p = 3 S 3 p = k S k

R2 3 3 R3 4 4

R2 2

R3

3

Rn

n

n Rn

n Rn

Rn

e1 = (1, 0, 0, . . . , 0, 0) Rne2 = (0, 1, 0, . . . , 0, 0) Rn

=en = (0, 0, 0, . . . , 0, 1) Rn

Rn

e1, e2, . . . , en Rn

Rn = {e1, e2, . . . , en}

S S

R4

e1 = (1, 0, 0, 0) e2 = (0, 1, 0, 0) e3 = (0, 0, 1, 0) e4 = (0, 0, 0, 1) (a,b,c,d) R4 (a,b,c,d) = ae1 + be2 + ce3 + de4

{(1, 0), (1, 1), (0, 1)

} R2 (a, b) R2 (a, b) = a(1, 0)+0(1, 1) + b(0, 1)

R2 (1, 1) = (1, 0) + (0, 1)

{(1, 0), (1, 1)} R2

R2 (a, b) R2 (a, b) = a(1, 0) + (b a)(1, 1)

= {(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)} R4 (a,b,c,d) R4 (a,b,c,d) = a(1, 1, 1, 1) + (b a)(0, 1, 1, 1) + (c b a)(0, 0, 1, 1) + (d c b a)(0, 0, 0, 1)


32/265

v = {b1, b2, . . . , bn}

i

bi v v =ni=1

ibi

[v] =

12

n

.

Rn = {e1, e2, . . . , en} v = (v1, v2, . . . , vn)

v = v1e1 + v2e2 + + vnen [v] =

v1v2

vn

v = (2, 4) = {(1, 0), (0, 1)} = {(1, 1), (0, 1)} [v] =

24

[v] =

22

v = (2, 4)

= {(1, 1, 1), (0, 1, 1), (0, 0, 1)} v = (1, 2, 2)

[v] = 12

2

[v] =

110

= {(1, 1, 1), (0, 1, 1), (0, 0, 1)} v = (1, 2, 2) w = (1, 1, 0)

[w] = [v] =

110


33/265

v Rn

v = (1, . . . , n)

[v] = 1n

v =

1n

Rn n

x = 4 R2 x = 1 R y = 3 R3 x + y = 2 R2 x y = 1 R3 x = 5y = 2 R2 x = 5y = 2 R3 x y = 5y = 2 R3

k > 1

k = 1

k < 1

k 1 < k < 0

(1, 2, 0, 0), (2, 4, 0, 0) + (2, 1, 2, 2) (1, 2, 0, 0), (0, 1, 0, 0) + (0, 0, 0, 0)

(1, 1, 1, 1) + (0, 0, 0, 0) (0, 0, 0, 0) + (1, 1, 1, 1)

u

v

w

u

w

v

w

S 5 Rn n = 3 S n = 4 S R4


34/265

R5 = { }

= {w1, w2, w3, w4} R4 u = w4 + 2w3 + 3w2 + 4w1

[u] =

S

S S S S

u = (2, 3) v = (1, 4) w = (2, 1)

u + 2v u v 3u 2v + w

(1, 2, 3, 2, 1) + (1, 2, 3, 4, 0) =

3(1, 2, 3, 2, 1) =

R2

y 2x = 5 y = 1

R3

z x = 1

x + y + z = 0

x + y = 1x y = 1

x = yz = 0

R3

(2, 3, 1)

(1, 2, 1)

(1, 2, 1) (0, 0, 1) x y = z 1 3x y + 1 = z

R3

x + y z = 2 y z = 0

R3

(1, 0, 1) (0, 1, 1) (1, 0, 0) (3, 0, 1) (2, 1, 1)

(0, 1, 1) (1, 3, 2) (

1, 2, 1) (1,

1,

1)

(3, 1, 0)

x = t + 1y = 1 tz = t 1

r = (1, 2, 0, 0) + t(0, 1/2, 1, 1) r (1, 4, 4, 4) r (1, 4, 3, 2) r r = (1, 4, 3, 2) + s(0, 1/2, 1, 1); r = (1, 4, 4, 4) + s(0, 2, 4, 4).

= (1, 1, 2, 0) + t(1, 2, 1, 2) + s(1, 1, 1, 1) R4


35/265

(2, 5, 3, 4) (1, 1, 3, 3) = (1, 1, 3, 3) + (1, 2, 1, 2), (1, 1, 1, 1)

{(x, y, z, w) R4| x y + 3z 2w = 4} {(x, y, z, w, u) R5| z 3u = 5}

{(1, 2, 2, 3), (2, 4, 4, 5)} {(1, 2, 1, 3), (3, 6, 3, 9} {(1, 2), (2, 1), (3, 3)} {(1, 2, 3, 4, 5), (0, 0, 0, 0, 0), (5, 4, 3, 2, 1)}

(1, 2, 3, 5) (1, 2, 3, 4) (1, 0, 0) (2, 1, 1), (3, 1, 1) (1, 0, 2) (2, 1, 1), (3, 1, 1) R3 = (0, 1, 0), (1, 0, 1), (0, 0, 1)

(2, 1, 2) = (2, 1, 2)

v = (4, 1, 1) = {(1, 1, 0), (0, 1, 1), (0, 0, 1)} v [v] [v]

[w] =

23

2

[w]

S 5 Rn n = 7 S n = 3 S R3

x = 3 R2

2x 3y + 5z = 1

x + y = 1

R3

x 2y = 1 R3 3x 2z 5 = 0 R3

(1, 2, 1, 2, 1) + (0, 0, 0, 0, 0), (1, 2, 1, 2, 1))

(1, 2, 1, 1) + (1, 2, 1, 3), (1, 2, 1, 4)) (1, 2, 1, 1) + (1, 1, 1, 1), (0, 2, 0, 2), (1, 3, 1, 3)) (2, 0, 2, 0) + (1, 2, 0, 0), (1, 1, 1, 0), (0, 0, 0, 0) (0, 0, 0, 0) + (0, 0, 0, 0))

v + u, u, 3u u = 0


36/265


37/265


38/265

x + y = 100

10x + 20y = 1250.

a b c d a 3 8 + b 2 c 2 + d 2

3a = c

8a = 2d

2b = 2c + d

3a +0b 1c +0d = 08a +0b +0c 2d = 00a +2b 2c 1d = 0

.

y = ax2 + bx + c (0, 1) (1, 3) (2, 4) (3, 9)

(0, 1) 1 = a(02) + b(0) + c(1, 3) 3 = a(12) + b(1) + c(2, 4) 4 = a(22) + b(2) + c(3, 9) 9 = a(32) + b(3) + c

4 a,b,c

0a +0b +1c = 11a +1b +1c = 34a +2b +1c = 49a +3b +1c = 9

.

f(x) = ax3 + bx2 + cx + d cos(x) ki i = 1, . . . , N N

ak31 +bk21 +ck1 +d = cos(k1)

ak3N +bk2N +ckN +d = cos(kN)

.


39/265

a (20 + 25 + b + d)/4

(a,b,c,d,e,f)

4a

b

d = 454b a c e = 154c b f = 254d e a = 554e b d f = 204f c e = 35

.

25o

15o

30o

20o

20o 25o

15o 20o10

o

15o

a

b

c

d

e

f

4

5 100 100 10 100100100 1

Rn n

N (xi, yi)


40/265

a, b R y = ax + b (xi, yi)

a, b N

ax1 + b = y1

=

axN + b = yN.

x

y

(0, 6, 10) (1, 2, 3) (2, 1, 1) (4, 1, 3)

, ,

(1, 2, 3) + (2, 1, 1) + (4, 1, 3)= (, 2, 3) + (2 , , ) + (4, , 3)= ( + 2+ 4, 2 + , 3 + 3)= (0, 6, 10).

1 +2 +4 = 02 +1 1 = 63 +1 3 = 10

R

R2 Rn

R3


41/265

R

1 1 x R

ax = b .

a = 0 x = a1b

a = b = 0 x R

a = 0 b = 0 x R

2 2 3 3 Ax = b b R2 R3 det(A) = 0 a = 0

x = A1

b

A

b

R2

a11x + a12y = b1 (r1)a21x + a22y = b2 (r2),

r1 r2

x

a11a21

+ y

a12a22

=

b1b2

.

v1 =

a11a21

, v2 =

a12a22

, b =

b1b2

,

b

v1 v2

x, y R xv1 + yv2 = b.

1x +1y = 2 (r1)1x 1y = 0 (r2) .


42/265

v1 =

11

, v2 =

1

1

, b =

20

.

(1, 1)

b

v1 v2 b = 1v1 + 1v2

x

y

(2, 0)r1

r2

(0, 2)

(0, 0)

(1, 1)x

y

v1

v2

b

1x 2y = 2 (r1)

2x +4y = 2 (r2) v1 =

1

2

, v2 =

24

, b =

22

.

b

v1 v2

x

y

r1

(1, 0)(0, 1/2)

r2

(2, 0)

(0, 1)x

y

v1

v2

b

1x 2y = 2 (r1)

2x +4y = 4 (r2)

v1 =

1

2

, v2 =

24

, b =

2

4

.

b


43/265

v1 v2 b =0v1 v2 = 2v1 + 0v2 = v1 1/2v2 {(x, y) | x + y =2} = {(t, 2t) = (0, 2) + t(1, 1) | t R} (0, 2) + (1, 1)

x

y

r1 = r2(2, 0)

(0, 1)

x

y

v1

v2

b

5


44/265

m n

a11x1 +a12x2 +a1nxn = b1a21x1 +a22x2 +a2nxn = b2

am1x1 +am2x2 +amnxn = bm

a11 a12 a1na21 a22 a2n

m1 am2 amn

b1b2

bm

m n m n

A A

A aij = 0 i = j.

1 00 3 ,

10 0 00 3 0

0 0 5

,

3 0 0 00 5 0 00 0 5 00 0 0 3

.

3x1 = 5

2x2 = 4x3 = 2

3 0 0 50 2 0 40 0 1 2

5

3, 2, 2


45/265

A aij = 0 i > j.

5 20 3

,

1 1 70 3 2

0 0 1

,

3 12 0 30 5 3 00 0 5

1

0 0 0 3

.

3x1 +x2 +3x3 = 2

2x2 +x3 = 52x3 = 2

3 1 3 20 2 1 50 0 2 2

. x3 x2 x1 x2 x1

2x3 = 2 x3 = 12x2 +(1) = 5 x2 = 2

3x1 +(2) +3(1) = 2 x1 = 1.

{(1, 1)}


46/265

1 1 21 1 0

1 2 31 1 03 1 4

(2, 0)

(0, 2)

(1, 1)

(3, 0)

0,

3

2

(1, 1)

43

, 0

l1 l2 l1 b1l2 b2

l2 b2l1 b1

;

l2 l2 l1 b1l2 b2

l1 b1l2 b2

;

l2 l2 + l1 l1 b1l2 b2

l1 b1l2 + l1 b2 + b1

;

l1 b1

0 0 0 0

l1 b1 .

A B A B

A

B

A B

A = BC = D

C = DA = B


47/265

A = BC = D

A = B

C = D C = D C = D

= 0 C = D = 0 1 1C = 1D 1 = 1 C = D

A = BC = D

A = BC+ A = D + B

C = D A = B A = B

= 0 C = D C + A = D + B C + A = D + B A = B A = B A C + A = D + B C = D + B A = D

A = B0 = 0

A = B

0 = 0

B = CB = C

B = C

B B = C C = 1

B = C0 = 0

B = C

A = BC = D

A + C = B + DA + C = B + D

A + C = B + D

x + y = 3

2x y = 0

{(1, 2)} 3x = 33x = 3

{(x, y) =(1, t); t R}

3x = 3

2x y = 0

3x = 3

5x y = 3 {(1, 2)}


48/265

x + y + z = 3

y + z = 2x + y = 2

x = y = z =

1

1 1 1 30 1 1 21 1 0 2

. l1

l1

l2 l2

l2

l3 l3

l3

l1

1 0 0 1

1 0 1 00 0 1 1 .

x = 1

x + z = 0z = 1

x = z = 1 y

{(x, y, z ) = (1, t, 1) = (1, 0, 1) + t(0, 1, 0), t R} (1, 0, 1) + (0, 1, 0)

1 2 4 20 7 11 253 13 4 16

. l3 l3 + 3l1

3 13 4 16+ 3 ( 1 2 4 2 ) =

3 13 4 16+ 3 6 12 6

0 7 8 22

1 2 4 20 7 11 250 7 8 22

. l3 l3 + l2 0 7 11 25+ 0 7 8 220 0 3 3

1 2 4 20 7 11 250 0 3 3

. l3 13

l3

1 2 4 20 7 11 250 0 1 1

. l1 l1 + 4l3

1 2 4 2+ 0 0 4 4

1 2 0 2. l2 l2

11l3 0 7 11 25

+ 0 0 11 110 7 0 14

.

1 2 0 20 7 0 140 0 1 1

.

l2 17

l2

1 2 0 20 1 0 20 0 1 1

. l1 l1 + 2l2 1 2 0 2

+ 0 2 0 41 0 0 2

1 0 0 20 1 0 20 0 1 1

. l1 l1

1 0 0 20 1 0 20 0 1 1

.


49/265

x1 = 2x2 = 2x3 = 1

.

{(2, 2, 1)}

4 7 0 14 40 0 4 0 10 0 0 13 6

4 7 0 14 43 0 4 0 1

0 0 0 13 6

4, 5, 13

4 7 0 14 40 0 5 0 10 0 0 13 6


50/265

1 7 0 0 40 0 1 0 10 0 0 1 6

1 7 2 0 40 0 1 0 10 0 0 1 6

p k 1

k < p

lk, lk+1, . . . , lp

lk+1, . . . , lp lk

p k k + 1

k = p, p 1, . . . , 1

lk

l1, . . . , lk1 lk

+

2 6 3 1 42 6 3 2 10

4 12 7 0 106 18 11 0 14

.


51/265

p 4 k 1

l1 l2 l3 l4

2 6 3 1 42 6 3

2 10

4 12 7 0 106 18 11 0 14

.

2 l2, l3, l4 l1

l2 l2 l1 l3 l3 + 2l1 l4 l4 3l1

2 6 3 1 40 0 0 3 60 0 1 2 20 0 2 3 2

p 4

k 2

2 3 4

2 6 3 1 40 0 0 3 60 0 1 2 20 0 2 3 2

.

2 6 3 1 4

0 0 0

3 6

0 0 1 2 20 0 2 3 2

2 6 3 1 4

0 0 1 2 20 0 0 3 60 0 2 3 2

1 l3 l4 l2

l4 l4+2l2 l3

2 6 3 1 4

0 0 1 2 20 0 0

3 6

0 0 0 1 2

p 4 k 3

3 4

2 6 3 1 40 0 1 2 20 0 0 3 60 0 0 1 2


52/265

2 6 3 1 40 0 1 2 20 0 0 3 60 0 0 1 2

3 l4 l3

l4 l4 + 1/3l3

2 6 3 1 40 0 1 2 20 0 0 3 60 0 0 0 0

2 6 3 1 40 0 1 2 20 0 0 3 6

p 3 k 4 k p

k 3

l3 3

2 6 3 1 40 0 1 2 20 0 0 1 2

l1, l2 l3 l1 l1 l3

l2 l2 2l1 2 6 3 0 60 0 1 0 20 0 0 1 2

k 2

l2 1

2 6 3 0 60 0 1 0 20 0 0 1 2

l1 l2 l1 l1 3l2

2 6 0 0 12

0 01

0 20 0 0 1 2 k 1

l1 2

1 3 0 0 60 0 1 0 20 0 0 1 2

1

k = 1


53/265

0 1

0 0 0 1

0x1 + 0x2 + + 0xn = 1 0 = 1

1 0 0 0 1 0

0 0 1

x1 = x2 =

xn =

{( , , . . . , )}

1 0 00 1 00 0 1

1 3 0 5 00 0 1 2 00 0 0 0 1

1 0 0 20 1 0 00 0 1 11

1 0 0 0 70 1 0 0 40 0 1 0 30 0 0 1 13

1 3 0 5 0 40 0 1 2 0 00 0 0 0 1 2

. x2 x4 x2 = r x4 = s

1x1 = 4 + 3r 5s

1x3 = 2s1x5 = 2

x1 x3 x5

1 0 0 4 + 3r 5s0 1 0 2s0 0 1 2


54/265

(x1, x3, x5) = (4 + 3r 5s, 2s, 2) x2 = r x4 = s (x1, x2, x3, x4, x5) = {(4 + 3r 5s, r, 2s, s, 2) | r, s R}

{(4, 0, 0, 0, 2) + r(3, 1, 0, 0, 0) + s(5, 0, 2, 1, 0) | r, s R}.

(4, 0, 0, 0, 2) + (3, 1, 0, 0, 0), (5, 0, 2, 1, 0) . 5 3

5 3 = 2 x2 x4 r, s 3

n + 1 n p

n p n p t , r , s , . . .

n p n p

0 1 0 0 0 1

0 1 2 0 0 0 1 00 0 0 1 0 0 3 40 0 0 0 1 0 0 10 0 0 0 0 1 3 2

.


55/265

n = 7 p = 4 7 4 = 3 1, 3, 7 x1, x3, x7 4 x2, x4, x5, x6 r,s,t x1 = r, x3 = s, x7 = t x2 = 2x3 + x7 = 2s + t, x4 = 4 3x7 = 4 3t, x5 = 1, x6 = 2 3x7 =2 3t (x1, x2, x3, x4, x5, x6, x7) = (r, 2s + t, s, 4 3t, 1, 2 3t, t)

{(0, 0, 0, 4,

1, 2, 0) + r(1, 0, 0, 0, 0, 0, 0) + s(0,

2, 1, 0, 0, 0, 0) + t(0, 1, 0,

3, 0,

3, 1)

},

r,s,t R (0, 0, 0, 4, 1, 2, 0) + (1, 0, 0, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0, 0), (0, 1, 0, 3, 0, 3, 1) .

0 1 3 0 70 0 0 1 4

.

n = 4 p = 2 4 2 = 2 1, 3 x1 x3 2 x2, x4 r, s

x1 = r

x3 = s

x2 = 73x3 = 73s x4 = 4 (x1, x2, x3, x4) = (r, 73s, s, 4)

{(0, 7, 0, 4) + r(1, 0, 0, 0) + s(0, 3, 1, 0) | r, s R} (0, 7, 0, 4) + (1, 0, 0, 0), (0, 3, 1, 0)

r, s r = 0 s = 0 (0, 7, 0, 4) + 0(1, 0, 0, 0) +0(0, 3, 1, 0) = (0, 7, 0, 4). r = 3 s = 2 (0, 7, 0, 4)+3(1, 0, 0, 0) 2(0, 3, 1, 0) = (3, 1, 2, 4). r s

S

S = {v0 + t1v1 + + tqvq, vi Rn, ti R},

S = v0 + v1, . . . , vq .

S q = n p v0, v1, v2, . . . , vq


56/265

0 0 0 1

1 0 0 0 1 0

0 0 1

0 0 0 0 0 0 0 0

0 1 0 00 0 0 1 0

0 0 0 0 1

0

0 0 0

1 0 0 0 1 0

0 0 0 1

0 0 0 0 0 0

1 0 0 0 0 1 0 0

0 0 1 0 0 0 0 1

0 0 0

0

0 0


57/265

0 3 0 1 60 0 0 90 0 0 0 311

0 = 311

13 2 0 6 330 107 2 9 10 0 0 3 0

1

2 2 8 12 00 e3 11 1 1

2

0 0 log(3) 2 00 0 0 77

3

2, e3

log(3) 77

u = (u1, . . . , un), v =(v1, . . . , vn) Rn u v

u v =ni=1

uivi.

u = (1, 2, 3, 4, 5), v = (1, 2, 1, 3, 0) R5 u v = (1)(1 ) + (2)(2)+(3)(1) + (4)(3) + (5)(0) = (1 ) + (4)+ (3) + (12) = 10.

a11 x1 + a12 x2 + a1n xn = b1a21 x1 + a22 x2

+ a2n xn = b2

am1 x1 + am2 x2 + amn xn = bm.

x =

x1xn

b = b1

bm

A =

a11 a12 a1na21 a22 a2n

am1 am2 amn

Ax = b.


58/265

u1 =

a11a12

a1n

u2 =

a21a22

a2n

uj = a

j1aj2

ajn

uj

A =

u1 um

x = (x1, . . . , xn) A

u1 x = b1u2 x = b2

um x = bm.

Ax =

u1

um

x

=

=

u1 x

um

x

=

b1

bm

= b.

a11 x1 + a12 x2 + a1n xn = b1a21 x1 + a22 x2 + a2n xn = b2

am1 x1 + am2 x2 + amn xn = bm

v1 vn b

x1

a11a21

am1

v1

+x2

a12a22

am2

v2

+ + xn

a1na2n

amn

vn

=

b1b2

bm

b

,

nj=1

xjvj = b. A =

v1

vn


59/265

Ax =

v1

vn

x1

xn

=

= nj=1

xjvj =

= b

Rn

n 1

R3 3 1 = 2 R2

2 1 = 1 R4 4 1 = 3

(x1, . . . , xn) Rn

n 1 n

1

(x, y, z, w, u) R5 x 2y + 3z+ w u = 4 x = 2y 3z w + u + 4 t1 = y, t2 =z, t3 = w t4 = u x = 2t1 3t2 t3 + t4 + 4 (t1, t2, t3, t4) (x, y, z, w, u) =(4, 0, 0, 0, 0) + t1(2, 1, 0, 0, 0) + t2(3, 0, 1, 0, 0) + t3(1, 0, 0, 1, 0) + t4(1, 0, 0, 0, 1) 4 R5 R5

A =

v1

vn A = u1

um n m Ax

uj

x = bj Hj n 1 Rn

mj=1

Hj

R2

R2

R3

R3

R3

4 R4

3


60/265

b

b =

xjvj xj R b v1, . . . , vn

b

b v1, . . . , vn

b =

xjvj

Hj =

a11x1 +a12x2 +a1nxn = 0a21x1 +a22x2 +a2nxn = 0

am1x1 +am2x2 +amnxn = 0.

(0, 0, . . . , 0)

0 0

0

0

. 0 0

n p

p = n p < n (n p)

Ax = b S

v0 S


61/265

0 1 3 0 70 0 0 1 4

.

{(0, 7, 0, 4) + r(1, 0, 0, 0) + s(0, 3, 1, 0) | r, s R}

(0, 7, 0, 4) + (1, 0, 0, 0), (0, 3, 1, 0) .

0 1 3 0 00 0 0 1 0

.

{r(1, 0, 0, 0) + s(0, 3, 1, 0) | r, s R}

(1, 0, 0, 0), (0, 3, 1, 0) .

(0, 7, 0, 4)

=

v0

S

V

v0 S S = v0 + V

S = Ax = b v0 S

V Ax = 0 S = v0 + V v0 + V S

S v0 + V

v V A(v0 + v) = Av0 + Av = b + 0 = b v0 + V S

w

S A(w

v0) = Aw

Av0 = b

b = 0

w v0 V w v0 + V S w0 + V

Ax = b v0+v Ax = 0

V S


62/265

S

S

v0

V

x + 2y = 4

2x + 5y = 9

x + 2y = 1

2x + 5y = 3

1 2 42 5 9

1 2 40 1 1

1 0 20 1 1 ,

x = 2, y = 1 1 2 12 5 3

1 2 10 1 1

1 0 10 1 1

,

x = 1, y = 1

1 2 4 12 5 9

3

1 2 4 10 1 1

1

1 0 2 10 1 1

1

x + y = 1

2x + 2y = 2

x + y = 1

2x 3y = 2

x + y = 1

2x + 2y = 3

1


63/265

0

x 2y = 33x + y = 1

2x + 3y =

ABC

ABC

x + y = 32x 2y =

x + y = 32x + y =

1 4 6 20 2 5 20 0 3 10 0 0 0

1 4 6 20 2 5 20 0 3 10 0 0 1

0 1 0 20 0 1 2

R5

Ax =

u1 um

x

= v1

vn

x1

xn

= b. Hj = {x Rn| uj x = bj}

Ax = b mj=1

Hj =,=


64/265

Ax = b b , v1, . . . , vn S = V

S = v0 + V v0 V S = v0 + V v0 S V = v0 + S v0 S V = v0 + S v0 V

x + 3y = 12x + y = 3

2x + 2y = 0

x y

x y

R2

x + y = 3x y = 1

3x y = 6

6x + 2y = 6

R3

ABCD AB CD r,s,t,u r A B s B C t A C u A D

st

us

rst

rtu


65/265

1 2 3 44 5 6 76 7 8 9

1 3 5 73 5 7 95 7 9 1

0 1 2 0 00 0 0 1 00 0 0 0 1

0 1 2 0 00 0 0 1 0

1 0 0 10 1 0 30 0 1 2

3

4 2 0

9 12 6 06 8 4 0

0 0 0 0

2 6 3 1 42 6 3 2 10

4 12 7 0 106 18 11 0 14

0 1 2 1 1 60 2 4 2 4 18

0 1 2 2 3 13

.

2 1 2 41 0 0

1 0 2 24 0 01 1 0 17 13

0 1 0 2 24 0 00 1 2 7 0 00 0 0 0 1 0 .

2x y + 2z = 1

4x + 2y 4z = 2

x + y 2z = 2x y = 0

2x + y 3z = 3

m 2x + 8y + 2z = 34x + m2y + mz = 6

a,b,c,d R x + 2y + 3z = a

5y + 6z = bcz = d

y(x) = ax2 + bx + c (1, 2) (2, 4) (3, 8) a,b,c R 6x + 3y = 9

4x + 3y = 5

6x + 3y = 34x + 3y = 1

6 3 9 34 3 5 1

.

1 0 12 1 3

0 4 3

21

1


66/265

A =

1 4 52 5 73 6 9

x Ax = 0

A P P A

A B B A

n

n

n n

R

2

2x 3y = 1

6x + 9y = 3

2x + 2y = 6x y = 1

x + 3y = 6

1 6

2 6

3 4

1 3 5

3 5 6

1 3 6


67/265

R2 (A, B) r

(x A)2 + (y B)2 = r2 a,b,c A,B,r

x2 + ax + y2 + by + c = 0 (x1, y1), (x2, y2), (x3, y3)

a,b,c

(4, 5), (2, 7) (4, 3) ,

x + 2y = 02x + y = 2

x + y + z = 1

2x y + 3z = x + 3y = 2

x + 2y + (+ 1)z = 2y + 2z = + 1

x + (1 )z = 0

x y + z = 1x + y z = 1

3x y + z = 1x + y + ( 1)z =

1 1 3 21 2 4 31 3

x + y

z = 2

y + 5z = 5x + 2y + z = 7

a x y = a

x + y + z = a2x + z = a

b1 b2 b3

2 5 8 b12 1 0 b21

4 6 b3

y = 4x4 + 3x

3 + 2x2 + 1x + 0

(a1, b1) (a2, b2) (a3, b3) (a4, b4) (a5, b5) 4, 3, 2, 1, 0 (ai, bi)


68/265


69/265

Rn

R

V

u, v, w V

u + v = v + u

(u + v) + w = u + (v + w) u, v, w

0

u + 0 = u

u

u

(u)

u + (u) = 0

u V , ()u = (u) , u

1u = u u


70/265


71/265

R2

R3

f F({1, 2} ;R) f : {1, 2} R f(1) f(2) f F({1, 2} ;R) f = (f(1), f(2)) R2 (a1, a2) R2 f F({1, 2} ;R) f(1) = a1 f(2) = a2 f = (5, 3) R2 f

F({

1, 2}

;R) g F({1, 2, 3} ;R) g : {1, 2, 3} R g = (g(1), g(2), g(3)) R3 (a1, a2, a3) R3 g F({1, 2, 3} ;R) g(1) = a1 g(2) = a2 g(3) = a3 g = (3, 5, 2) R3 g

x

y

1

f(1) = 5

2

f(2) = 3

x

y

1

g(1) = 3

2

g(2) = 5

3

g(3) = 2

f = (5, 3) R2 g = (3, 5, 2) R3

f = (2, 4, 3, 4, 1) R5 f F({1, 2, 3, 4, 5} ;R) f(i) = ai i = 1, . . . , 5

Rn n > 3

x

y

1

f(1) = 2

2

f(4) = f(2) = 4

3

f(3) = 3

4 5

f(5) = 1

f = (2, 4, 3, 4, 1) R5

n f = (a1, . . . , an) Rn f : {1, . . . , n} R f(1) = a1, f(2) =a2, . . . , f (n) = an f Rn n f F({1, . . . , n} ;R) {1, . . . , n} I I R f F(I;R) f : I R f F([0, ];R) f(x) = sen(x)


72/265

x

y

0

f F([0, ];R) f(x) = sen(x)

In = {1, . . . , n} Rn

u, v Rn u, v : In R u + v : In R (u + v)(i) = u(i) + v(i) i = 1, . . . , n

f, g F([0, 1] [0, 1];R) [0, 1] [0, 1] r(t) = tg + (1 t)f r(0.0) = f r(1.0) = g f = r(0.0) g = r(1.0) r r(0.2), . . . , r(0.8)

r(0.0) r(0.2) r(0.4) r(0.5) r(0.6) r(0.8) r(1.0)

V

{0} V

V V V


73/265

{0} V V

H V 0 H H u, v H u + v H

H u H, R u H

u V H = {v V| v = u, R} V

v

0 H v1, v2 V v1 = 1u, v2 = 2u v1 + v2 = (1 + 2)u H R v1 = (1)u1 H

H = {(x,y, 0), x,y R} R3

(0, 0, 0) H

(x1, y1, 0), (x2, y2, 0) H (x1, y1, 0)+(x2, y2, 0) = (x1+x2, y1+y2, 0) H

(x,y, 0) H R (x,y, 0) = (x, y, 0) H

H = {(x,y, 1), x,y R} R3 (0, 0, 1)

(0, 0, 0) H

(x1, y1, 1), (x2, y2, 1) H (x1, y1, 1)+(x2, y2, 1) = (x1+x2, y1+y2, 2) H

(x,y, 1) H R (x,y, 1) = (x, y, ) H = 1

r 0 r H = {(x, x2). x R} R2 (0, 0) H (1, 1), (2, 4) H (1, 1) + (2, 4) = (3, 5) H H R2 v Hv H H = {p P3| p(1) = 0} P3

0 H p, q H p(1) = q(1) = 0

(p + q)(1) = p(1) + q(1) = 0

p H R p(1) = 0 (p)(1) = (p(1)) = (0) = 0


74/265

V H V H H W V h0 V W V H =h0 + W = {h0 + w | w W}

A Mnm

v1, v2, . . . , vn Rm A = v1

vn

V = {x Rn| Ax = 0} Rn Ax = 0

W = {b Rm| b = Ax; x Rn} W =

ni=1

xivi| xi R

W =

v1, v2, . . . , vn

V W

A A Ax = 0 A

F(I;R)

Pn

F(I;R) I R Pn F(I;R)

f F(I;R) p Pn f

I

C(I;R) I R Ck(I;R) k

C(I;R)


75/265

f(x) = cos(x), g(x) = exp(x2) C(R;R) f(x) = 1/x g(x) = 1/(x21) C((0, 1);R)

F(R;R) 0 1 f(x) = |x| C(R;R) C1(R;R)

0

k k 1

Pn P C(I;R) Ck(I;R) C2(I;R) C1(I;R) C(I;R) F(I;R).

V = {y(x)| y(x) + 9y(x) = 0} F(R;R)V y

1(x), y

2(x)

V y =

ay1 + by2 a, b R y+9y = ay1+by

2+9(y1+y2) = ay

1+9y1+by

2 +9y2 =

0 + 0 = 0

y1(x) = sen(3x) y2 = cos(3x) y1, y2 V

V = {a sen(3x) + b cos(3x); a, b R}

V = {y C(R;R)| y(x) + 9y(x) = 9x} y1(x), y2(x) V y = y1 + y2 y + 9y = 9x + 9x = 18x = 9x

y

V y0(x) = x y0 V V = y0 + {a sen(3x) + b cos(3x); a, b R}

V = {y C(R;R)| y(x) + f(x)y(x) = 0} y1(x), y2(x) V

y = ay1 + by2 a, b R y + f y = ay1 + by

2 + f(y1 + y2) =

ay1 + f y1 + by2 + f y2 = 0 + 0 = 0 y V

V = {y C(R;R)| y(x) + y2

(x) = 0} y w = ay

w + w2 = ay + a2y2 = ay + ay2 + ay2 = 0 + ay2 = 0 V

Rn


76/265

v

v1, v2, . . . , vp v

v = 1v1 + 2v2 + + pvp =pi=1

ivi,

i

(2, 1, 7) (1, 2, 3) (4, 5, 6) (7, 8, 7) , , R (1, 2, 3) + (4, 5, 6) + (7, 8, 7) =

(( + 4+ 7), (2 + 5+ 8), (3 + 6+ 7)) = (2, 1, 7)

1 +4 +7 = 22 +5 +8 = 13 +6 +7 = 7

1 4 7 22 5 8 13 6 7 7

1 4 7 20 3 6 3

0 0 2 7

1 0 0 5.50 1 0 8

0 0 1 3.5

= 5.5, = 8, = 3.5

P u = x3 + x,

v = x2 x, w1 = 3x3 x2 + 4x w2 = x3 + 2x2 + 10 w1, w2 u v

w1 w1 = 3x3 x2 + 4x = 3(x3 + x) (x2 x) = 3u v

w2 w2 = x3 + 2x2 +10 = (x3 + x) +

(x2 x) = x3 + x2 + ( 1)x , 10 w2 = u + v , R

F(R;R) u = sen2(x) v = cos2(x)

w = cos(2x)

u, v

w = v u

0 = 0v1 + 0v2 + + 0vp.

{v1, v2, . . . , vp} v1, v2, . . . , vp span {v1, v2, . . . , vp} v1, v2, . . . , vp

v1, v2, . . . , vp = span{v1, v2, . . . , vp} =

pi=1

ivi

i R, i = 1, 2, . . . , p

.


77/265

{v1, v2, . . . , vp} W W = v1, v2, . . . , vp

= {0}

{0}

0

i=1vi = 0

{v1, v2, . . . , vp}

H = v1, v2, . . . , vp u, v H u =pi=1

uivi, w =

pi=1

wivi

ui = 0 0 H u + w =pi=1

(ui + wi)vi H

R u =pi=1

(ui)vi H

(1, 0, 0), (0, 1, 0) = {(x,y, 0) | x, y R} = R2

H = {p P2| p(2) = p(3)}

P2 2 3

p(x) = ax2 + bx + c p(2) = p(3) 4a + 2b + c = 9a + 3b + c 5a + b = 0 c = r

b = s a = b/5 = s/5 V = {s/5x2 + sx + r; r, s R} 2 P2

r = 0, s = 1 u = x2/5 + x r = 1, s = 0 v = 1 V = u, v

H = {p P3| p(1) = 0} P3 p(x) = ax3 + bx2 + cx + d p(1) = 0 a + b + c + d = 0

d = r, c = s, b = t a = r s t H = {(r s t)x3 + tx2 + sx + r} r,s,t 0 1

u = x3 + 1, v = x3 + x, w = x3 + x2 H = u, v, w a = r, b = s, c = t

d = r s t

r,s,t

0

1

u= x

3

1,v

=x2 1, w = x 1 H = u, v, w

W = {p Pn | p(1) = 0}

W =

(x 1), x(x 1), . . . , xn1(x 1) . p W p(1) = 0 1 (x 1)

p(x) = (x 1)n1i=0

aixi

=n1i=0

ai

xi(x 1)


78/265

cos(2x) = 2 cos2(x) 1 cos(2x) 1, cos(x), cos2(x)

sen(2x) = 2sen(x)cos(x) cos(3x) = cos(x) cos(2x) sen(x) sen(2x) cos(3x) = 2 cos3(x) + 2 cos2(x) cos(x) 2

cos(3x) 1, cos(x), cos2(x), cos3(x) cos(nx) 1, cos(x), . . . , cosn(x)

0 1

2 3

0, . . . , 3

i(j) = ij ij 1 i = j 0 0(0) = 1 i = j

0(1) = 0(2) = 0 1(1) = 1 i = j 1(0) = 1(2) = 0

f f(j) = aj j = 0, . . . , 3

f =3i=0

aii f(0) =3i=0

aii(0) = a00(0) = a0 1 = a0

1(0) = 2(0) = 3(0) = 0 f(1) =

3i=0

aii(1) = a1

f(2) = a2 f(3) = a3 i

0, . . . , 3 = .

Rn


79/265

1, 2 3

2 3

vk

{v1, v2, . . . , vn} vk =pi=1i=k

ivi

1v1 + . . . + k1vk11vk + k+1vk+1 + . . . + pvp = 0. 0 vi

0

0

vi

pi=1

ivi = 0, k = 0.

(k) 1k

v1 . . . k1k vk11vk k+1k

vk+1 . . . pk vp = 0

vk =pi=1i=k

ik

vi.


80/265

v1, v2, . . . , vp

vk =i


81/265

1 +4 +7 = 02 +5 +8 = 03 +6 +7 = 0

1 4 7 02 5 8 03 6 7 0

1 4 7 00 3 6 0

0 0 2 0

(1, 2, 3) (4, 5, 6) (7, 8, 9) , , R (1, 2, 3) + (4, 5, 6) +

(7, 8, 9) = (( + 4+ 7), (2 + 5+ 8), (3 + 6+ 9)) = (0, 0, 0)

1 +4 +7 = 02 +5 +8 = 03 +6 +9 = 0

1 4 7 02 5 8 03 6 9 0

1 4 7 00 3 6 0

0 0 0 0

.

Rn

Rn

e1, e2, . . . , en v Rn v = (v1, . . . , vn) =ni=1

iei =

(1, 2, . . . , n) i = vi i = 1, . . . , n

S V V S

{e1, e2} R2 (a, b) R2 (a, b) = ae1 + be2

{(0, 1), (1, 0), (0, 1)} R2 (a, b) R2 (a, b) =a(1, 0) + (b + )(0, 1) + (0, 1) R R2 (3, 2) = 3(1, 0) + 2(0, 1) +0(0, 1) = 3(1, 0) + 1(0, 1) 1(0, 1)


82/265

= {(1, 1, . . . , 1), (0, 1, . . . , 1), . . . , (0, . . . , 0, 1)} = {b1, b2, . . . , bn} Rn

v = (v1, . . . , vn) Rn v =

ibi = (1, 1 + 2, 1 + 2 +

3, . . .)

1 = v11 + 2 = v21 + 2 + 3 = v3

1 = v12 = v2 1 = v2 v13 = v3

(1 + 2) = v3

v2

.

Rn

= {1, t, t2} P2 p P2 p(t) = at2 + bt + c1

= {1, t, t2, . . . , tn} Pn

n

i=0aiti =

n

i=0biti

t R ai =bi, i = 0, 1, . . . , n .

V V

V

0

v =n

i=0 ivi =n

i=0 ivi n

i=0 (i i)vi = 0 i i = 0 i = 0, . . . , n i = i i = 0, . . . , n

Rn

A =

u1 um

B = v1

vk

B

A u1, u2, . . . , um v1, v2, . . . , vk B

{v1, v2, . . . , vk}


83/265


84/265

v

[v] =

(v1, v2, . . . , vn)

=

v1

v2 v1

vn vn1

.

= {(1, 1, 1), (1, 0, 1), (2, 0, 1)} R3 v

[v] =

23

1

v = 2(1, 1, 1) + 3(1, 0, 1) (2, 0, 1) = (3, 2, 6)

= {(1, 1, 1), (1, 0, 1), (2, 0, 1)} R3

(4, 3, 7) [(4, 3, 7)] a1, a2, a3 R a1(1, 1, 1) + a2(1, 0, 1) + a3(2, 0, 1) =

(4, 3, 7) (a1 + a2 + 2a3, a1, a1 + a2 a3) = (4, 3, 7)

1a1 + 1a2 + 2a3 = 4

1a1 + 0a2 + 0a3 = 31a1 + 1a2 + (1)a3 = 7

1 1 2 41 0 0 31 1 1 7

1 0 0 30 1 0 3

0 0 1 1

(a1, a2, a3) = (3, 3, 1) [(4, 3, 7)] =

331

= {1 + x, 1 x, x2} u = 5 v = 6 + 5x2 w = 8x 2 [u] [v] [w]

5 = 5/2(1 + x) + 5/2(1 x) [u] = 5/25/2

0

6 + 5x2 = 3(1 + x) + 3(1 x) + 5x2 [v] =

335

8x 2 = (1 + x) + (1 x) = + + ( )x

+ = 8 = 2 = 3, = 5 [w] =

350


85/265

= {sen2(x), cos2(x)} = {1, sen2(x)} u = cos2(x) v = cos(2x)

w = 1 [u] [v] [w] [u] [v] [w]

cos(2x) = cos2(x) sen2(x) = 1 2sen2(x) [u] =

01

[v] =

11

[w] =

11

[u] =

1

1

[v] =

1

2

[w] =

10

Rn

V Rn

V

n [ ] : V Rnv [v]

[u + v] = [u]+ [v] , R, u, v V;

[v] = [w]

v = w

[u + v] = [u]+ [v] u, v V,

[u] = [u] R, u V.

= {v1, v2, . . . , vn} V u =n

i=1ivi, w =

n

i=1ivi, u +

w =ni=1

(i + i)vi [u] = 1

n

, [w] = 1n

, [u + w] = 1 + 1n + n

= 1

n

+ 1

n

= [u]+ [w]. [u] = 1

n

= [u]


86/265

Rn = {e1, e2, . . . , en}

Pn = {1, x , x2, . . . , xn}

P

= {p1, p2, . . . , pn} P N = maxp

grau(p)

q(x) = xN+1 q P q grau(q) = N + 1 > N grau(p)

p

C(I;R) Ck(I;R) C2(I;R) C1(I;R) C(I;R) F(I;R)

P

= {u1, u2, . . . , um} , = {v1, v2, . . . , vn} H H m n

aij

vj =

m

i=1

aijui

A = [aij ] i=1,...,mj=1,...,n

n > m n m x = 0 Ax = 0

nj=1

xjaj = 0 nj=1

xjaij = 0 i

mi=1

nj=1

xjaij

ui = 0

nj=1

xj

mi=1

aijui

=

nj=1

xjvj = 0

n m

= {v1, v2, . . . , vm} = {u1, u2, . . . , un} m n n m m n n m m = n

S = {v1, v2, . . . , vn} vk v1, . . . , vk1, vk+1, . . . , vn = S .

vk =i=k

ivi. w S w =i

ivi =i=k

ivi +

kvk =i=k

ivi + ki=k

ivi =i=k

(i + ki)vi w =i=k

(i + ki)vi

w v1, . . . , vk1, vk+1, . . . , vn


87/265

{v1, v2, . . . , vp} vp+1, . . . , vn {v1, v2, . . . , vp, vp+1, . . . , vn}

{v1, v2, . . . , vp} = {u1, u2, . . . , un} {v1, v2, . . . , vp, u1, u2, . . . , un}

vi vi

n

n

n

n

Rn

u v

u = v u v u v

u, v = u, w v = w v w v w

{w} w

{v1, v2, . . . , vn} 1 = = n = 0 1v1 + nvn = 0 1v1 + nvn = 0 i R 1v1 + nvn = 0 1 = = n = 0


88/265


89/265

R2

V W R3

V

W V + W

(1, 0, 6) (1, 0, 1), (1, 2, 1) (1, 2, 1) (1, 0, 1), (1, 2, 1)

{(0, 0, 2, 2), (3, 3, 0, 0), (1, 1, 0, 1)}

{(0, 0, 0)} R3

{(1, 0, 2)} R3

{(0, 0, 0), (1, 0, 2)} R3 {(1, 0, 2), (1, 2, 1)} R3 {(1, 0, 2), (2, 0, 4)} R3

{(1, 0, 2), (2, 1, 1), (4, 1, 1)} R3

{(1, 0, 2), (2, 1, 1), (1, 2, 8)} R3

R3

11

1

, 12

3

11

1

, 00

0

, 30

1

111

,

123

,

301

,

012

11

1

, 222

, 301

R4

{(x, y, z, w) | x + y w = 0}

x + z w = 0

z+ w = 0

0 1 3 0 00 0 0 1 0


90/265

r + 2s + ts 2t

2r + s 4tr 3t

r,s,t R

x y = 0y + w = 0x + w = 0

v = (0, 5, 1) [v] v = {(1, 1, 1), (1, 1, 0), (1, 0, 1)}

u, v, w V {u v, v w, w u}

{v1, v2, . . . , vn} W = v1, v2, . . . , vn

w W {w, v1, v2, . . . , vn}

{v2, . . . , vn}

v2, . . . , vn = W

F([a, b];R) f : [a, b] R

f(a) = f(b) = 0 f(a) = f(b) = 1

f(x) 0 x [a, b] f f + 2f = 0

{1 + 2x, 1 + x, 1 x} P2

{1, sen2(x), cos2(x)} F(R;R)

P3

{1, x2, x2 + 4} {x + 2, x + 3, x2 3} {x2 2x, 1 + x, x2 + 2} f : [a, b] R

C(R;R) {f C(R;R)| f = 0} g g(x) = 1

{f C(R;R)| f f = 0} g g(x) = ex

P4

{p P4| p(2) = 0} {p P4| p(2) = 1} = {1, 1 x, x2 1}

[q] q(x) = x2 x [p] p(x) = x2 + x + 1

0, . . . , 3 = {0, . . . , 3} f : [0, 3] R


91/265

[f]

x

y

0

4

1

3

2

5

3

2

V R1 V = 0 V = R W V Rn dim(W) = dim(V) W = V

U V 3

R5 U V 1

H

W h0 V W V H = h0 + W ={h0 + w | w W}

H u, v H u + (1 )v H R

H, K V

H K

H+ K = {h + k | h H, k K}. H K = {0} H K

H K H K H+ K H + K H K W

H W K W H+ K W

dim(H+ K) = dim(H) + dim(K) dim(H K).

W1 = {(s + t, t s, s + t, 2

livro-ALGLIN.pdf

Documents

Transcript of livro-ALGLIN.pdf