Shmeiwseis Gram & Ypol Algebra 2005 - 2006

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2005 2006 1 52

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9 : S={1,2,...,r} . 11+22+...+rr=0 (1) 1=2=...=r=0 (2) (2) (1) S . (2) S . 1. S={1,2,3} 1={2,-1,0,3} 2={1,2,5,-1} 3={7,-1,5,8} , 31+2-3=0 i=(1,0,0), j=(0,1,0) =(0,0,1) . S={1,2,3} 1={1,-2,3), 2={5,6,-1) 3={3,2,1) . .. 6 : S 1) . S S. 2) . S . 7 : 1) . 2) . . 8 : S={1,2,...,r} Rn.A r>n S .. 2 1. R . . ; 1={,- , - ), 2={-,, -), 3={- , - ,) 2. S={1,2,3} .. {1,2}, { 1,3}, {2,3} ..

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3. S={1,2,...,r} .. .. 4. S={1,2,...,r} .. S, S S .. 8 : V1=(U11,U12,...,U1n) V2=(U21,U22,...,U2n) ............................... Vr=(Ur1,Ur2 ,...,Urn ) 1V1+2V2+...+rVr=0 U11K1+U21K2++U r1Kr=0 U12K1+U22K2++U r2Kr=0 .. U1n K1+U2n K2++UrnKr=0

(1)

(2)

(3)

To (3) n r . r>n, (2) ., 1,2,...,r (2). S={1,2,...,r} ..

10 : V .. S={1,2,...,r} V. S V 1) S . 2) S V : S={e1,e2,...,en} e1=(1,0,0,...,0), e2=(0,1,0,...,0), ..., en=(0,0,0,...,1) Rn. Rn. S={1,2,3} 1=(1,2,1), 2=(2,9,0) 3=(3,3,4) R3. S={1,2,3,4} 3 52

. , 1= | 1 0 | 2= | 0 1 | 3= | 0 0 | |_ 0 0 _| |_ 0 0 _| |_ 1 0 _| .. 22 2x2 4= | 0 0 | |_ 0 1 _|

.

11 : .. V S={1,2,..., n} . V . .. . 9 : S={1,2,..., n} .. V, n .. : S={w1,w2,...,wm} m V m>n. S V w1=11 1+21 2+...+ n1 n w2=12 1+22 2+...+ n2 n.......................................................... (1)

wm=1m 1+2m 2+...+ nm n S .. 1,2,...,m K1W1+K2W2++KmWm=0 (2) (1) (2) 11 1+12 2+...+ 1m m=0 21 1+22 2+...+ 2m m=0 .................................................. (3) n1 1+n2 2+...+ nm m=0 3 n m m>n. . S .. 10 : S S .. V. S S . : S={1,2,..., n} S={w1,w2,...,wm} . .. V. S S . .9 mn (1) S S . .9 n m (2) m= n.

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12 : ... V . .. 0. : .. . 2x1+2x2+-x3 +x5=0 -x1-x2+2x3-3x4+x5=0 x1+x2-2x3 -x5=0 x3+x4+x5=0 : x1= -s-t , x2= s , x3 = - t , x4=0 , x5 = t | x1 | | -s-t | | -s | x2 | | s | | s | x3 | = | - t | = | 0 | x4 | | 0 | | 0 |_ x5 _| |_ t _| |_ 0 | -1 | 1 V1 = | 0 | 0 |_ 0 | | | V2= | _| | -1 | 0 | -1 | 0 |_ 1 | | | | _| | | - t | | -1 | | -1 | | | 0 | | 0 | | 0 | | + | - t | = s | -1 | + t | -1 | | | 0 | | 0 | | 0 | _| |_ t _| |_ 1 _| |_ 1 _|

. V1 V2. V1, V2 .. S={V1, V2} .. .. . 11 : .. V n. 1) S={1,2,..., n} n , S V. 2) S={1,2,..., n} V, S V. 3) S={1,2,...,r } . V r= 0, = 0 v=0 . : u=(u1, u2, un) v=(v1, v2, vn) Rn, =uv=u1v1 + u2v2 + + unvn n R . . Rn . Rn. . w1, w2, wn () u=(u1, u2, un), v=(v1, v2, vn) = w1u1v1 + w2u2v2 ++ wnunvn Rn. w1, w2, wn. : u=(u1,u2) v=(v1,v2) R2 = 3u1v1 + 2u2v2 . : 1. = 3u1v1 + 2u2v2 = 3v1u1 + 2v2u2 = 2. z = (z1,z2), = 3(u1+v1)z1 + 2(u2+v2)z2 = 3u1z1 + 3v1z1 + 2u2z2 + 2v2z2 = 3u1z1 + 2u2z2 + 3v1z1 + 2v2z2 = + 3. = 3(u1)v1 + 2(u2)v2 = (3u1v1) + (2u2v2) = (3u1v1 + 2u2v2) = 4. = 3v1v1 + 2v2v2 = 3v12 + 2v22

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v1, v2 R, 3v12 + 2v22 >= 0 v1=v2=0. Rn, . ut = [u1 u2 un], vt = [v1 v2 vn] nxn . uv Rn = AuAv Rn . uv vtu (2). (1) = (Av)tAu = vtAtAu (3) : Rn In . (1) A=In = IuIv = uv : () = 3u1v1 + 2u2v2 = 3 0 . : 0 2 = [v1 v2] 3 0 3 0 u1 = = 3u1v1 + 2u2v2 0 2 0 2 u2 , = w1u1v1 + + wnunvn Rn

. : U = u1 u2 V = v1 v2 u2v2 + u 3 u4 v 3 v4 + u4v4 22 . = u1v1 + + u3v3

: u,v,w R, :

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1. = = 0 2. = + 3. = (2): = = + = + = vtAtAu . : = (v+w)tAtAu = (vt + wt)AtAu = vtAtAu + wtAtAu = + : 1. = vtAtAu (1) ) = 9u1v1 + 4u2v2 R2 = 3 0 0 2 ) u=(-2,1) v=(2,9) = 9u1v1 + 4u2v2 ) (1) = 5u1v1 u1v2 u2v1 + 10u2v2 R2 = 2 1 -1 3 ) u=(0,-1) v=(1,4) 2. u=(u1,u2,u3) v=(v1,v2,v3) R3 : ) = u1v1 + u3v3 ) = u12v12 + u22v22 + u32v32 ) = 2u1v1 + u2v2 + 4u3v3 ) = u1v1 u2v2 + u3v3 3. U = u1 u2 u 3 u4 V = v1 v2 v 3 v4 = u1v1 + u2v3 + u3v2 + u4v4 M22 .

4. w1,w2,,wn u=(u1,u2,,un) , v=(v1,v2,,vn) Rn = w1u1v1 + w2u2v2 + + wnunvn Rn .

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& R2 u = (u1,u2) u = R3 u = (u1, u2, u3) u= u12 +u22= uu = (uu)1/2 uu = (uu )1/2

u12 +u22 + u32 =

.

: V u : u = < u , u >1/2

: V .. . . u v : d (u ,v) = u - v. : u = (u1, u2, , un) v = (v1, v2, , vn) Rn : u = < u, u >1/2 = u12 + u22 + + un2 d (u,v) = u v = < uv, uv >1/2 = (u1 1)2 + (u2 2)2 + + ( un n)2.

: . . . .. u = (1, 0) v = (0, 1) R2 12 + 02 = 1 d (u,v) = u v = ( 1, -1) = 12 + (-1)2 = 2 u= : < u, v > = 3 u1v1 + 2 u2v2

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: u= < u, u >1/2 = 3(1)(1) + 2(0)(0) = 3 d (u,v) = u v = < (1, -1), (1, -1) >1/2 = 3(1)(1) + 2(-1)(-1) = 5

: ( Cauchy Schwarz) V u, v V < u, v >2

: V u= < u, u >1/2 d (u,v) = u v : u 0 u = 0 u=0 k u = k u u + v= u + v (w,v) d (u,v) 0 d (u,v) = 0 u=v d (u,v) = d (v,u) d (u,v) d (u,w) + d

C S : u2 = < u, u > v2 = < v, v > : 2 2 2 u v (1) < u, v > u v u 0 v 0 : -1 uv

< u, v >

1

, 0 cos = < u, v > u v. :

uv

A u = (4, 3, 1, -2) v = (-2, 1, 2, 3) R4 cos = < u, v > = -8 +3 +2 -6 = -9uv

30 18

615

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: U = u1 u2 u3 u4 V = v1 v2 v3 v4

: < U, V > = u1v1 + u2v2 + u3v3 + u4v4 . M22. U= 1 0 1 1 V= 0 2 0 0 cos = < uv > =uv

U V = /2

0 +0 +0 +0

uv

= 0

: u v : < u, v > = 0. u W u W.

: ( ) u, v : u + v2 = u2 + v2 : u + v2 =

< (u+v) , (u+v) >

= u2 + 2< u, v > + v2

= u2 + 20 + v2 = u2 + v2

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y y bv2 P (a,b) v2av1+bv2

P

(a,b) b

O

a

x O v1 av1 x

v1 = v2 = 1 .

{v1,v2} R2. OP = av1 + bv2

a b P OP v1 v2 . v1 v2 1. R3 . v1 v2 = av1 + bv2 bv2 av1 + bv2

v2

v1

av1

(a, b) { v1 , v2 }.

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: S = { v1, v2, ,vn } V. v V v = c1v1 + c2v2 ++ cnvn . : v V v = c1v1 + c2v2 ++ cnvn (1) v = 1v1 + 2v2 ++ nvn (2) (1) (2) : 0 = (c1 - 1)v1 + (c2 2)v2 ++ (cn n)vn v1, v2, , vn . : c1 - 1 = c2 2 = = cn n = 0 : c1 = 1 , c2 = 2 , , cn = n

: S = { v1, v2, , vn } V v = c1v1 + c2v2 ++ cnvn c1, c2 , , cn S. S (v)s = ( c1, c2, , cn ) Rn . v S : c1 c2 [v]s = cn

1 : S = { v1, v2, v3 } v1 = (1,2,1), v2 = (2,9,0) v3 = (3,3,4). i. ii. iii. : S R3. v = (5, -1, 9) S. t v R3 [v]s = [-1 3 2] .

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. , i. ii.

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.... c1, c2, c3 R v = c1v1 + c2v2 + c3v3 : : (5, -1, 9) = c1(1, 2, 1) + c2(2, 9, 0) + c3(3, 3, 4) c1 + 2c2 + 3c3 = 5 2c1 + 9c2 + 3c3 = -1 c1 + 4c3 = 9 1 -1 2 => c1 = 1 c2 = -1 c3 = 2

[v]s = iii.

(v)s = (1, -1, 2)

v = -1v1 + 3v2 + 2v3 = -(1, 2, 1) + 3(2, 9, 0) + 2(3, 3, 4) = (-1, -2, -1) + (6, 27, 0) + (6, 6, 8) = (11, 31, 7).

: . . 2 : 1 ii v = c1v2 + c2v1 + c3v3 (5, -1, 9) = c1(2, 9, 0) + c2(1, 2, 1) + c3(3, 3, 4) 2c1 + c2 + 3c3 = 5 9c1 + 2c2 + 3c3 = -1 c2 + 4c3 = 9 [v]s = -1 1 2 => c1 = -1 c2 = 1 c3 = 2 (v)s = (-1, 1, 2)

3 : S = {v1, v2, , vn} V u V u = v1 + v2 + + vn (1) :

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. , (u)s = ( , , , ) [u]s =

. (2) (3)

v1 = (0, 1, 0), v2 = (-4/5, 0, 3/5), v3 = (3/5, 0, 4/5) R3 u = (2, -1, 4) = -1 , = 4/5 , = 22/5 (u)s = (-1, 4/5, 22/5) , -1 [u]s = 4/5 22/5

: S n- (u)s = (u1, u2, , un) , (v)s = (v1, v2, , vn) : i. u = u12 + u22 + + un2 ii. iii. d(u,v) = (u1 v1)2 + (u2 v2)2 + + (un vn)2

= u1v1 + u2v2 + + unvn

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. , T

.

= {u, u} .. V ' = {u', u'} .. V : [u'] = a b [u'] = c d ,

u' = au + bu

u' =cu + du (1) v V [v] = , v = ku' + ku' (3) (1) (3) : v = k(au + bu) + k (cu + du) v = (ka + kc) u + (kb + kd) u [v] = k1 k2 (2)

k1 a + k c2

a b

c d

k1 b + k 2 d

k1 k2

(4)

(2) (4) :

[v] =

a b

c d

[v]' (5)

(5) . [u'] [u'].

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={u, u,, un ) .. V '={u', u',, un }, [v] = P[v]' ' P = [[u'], [u'],, [u' ]]. n ' . : 1. = {u, u} & B' = {u', u'} u = (1,0), u = (0,1), u' = (1,1), u' = (2,1) . i) ii) : i) u' = u + u u' = 2u + u [u'] = 1 1 2 [u'] = 2 ' . [] []' = -3

5

.

1.

1

= ii)

11

[v] = P [v]' =

1 2

11

-3

5

=

7

2

: . : ) ) . : = 1 2

11

, = -1

2 . 1 -1-3 = 5 ..

v = (7,2) [v] =

7 2

[v]' =

-1 2 1 -1

7 2

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A: , , , :

= : : = .

: : i) . ii) t t . iii) .1/2 1/2 0 0 0 1 1/2 -1/2 0: : A =

: r = (1/2,1/2,0), r = (0,0,1) r = (12,-12,0). : r=r=r=1 rr = rr = rr = 0 r, r, r . t = . : 1. w S = {u,u}

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. , i) ii) u = (1,0), u = (0,1) w = (3,-7) u = (2,-4), u = (3,8) w = (1,1).

.

2. w S = {u,u} u = (1/2,-1/2), u = (1/2,1/2), w = (3,7) 3. = {u,u} = {v,v} u = i) ii) iii)

1 , u = 0

0 , v = 1

2 v = 1

-3 . 4

. . [w], w =

3 . -5

4. = {u,u,u} = {v,v,v} u = i) ii)

-3 -3 0 , u = 2 , u = -3 -1

1 -2 -6 -2 , v = , v = , v = 6 -6 -6 -3 -1 4 0 7

-5 [w], w = 8 . -5

5. : A=

0 1 ,B= 1 0

0 1 1/2 1 0 0 0 0 1/2 1/2 1/6 -5/6 1/6

1/2 1/2 1/2 1/2 -5/6 1/6 C= 1/2 1/6 1/6 1/2 1/6 -5/6

6. det(P) = 1. : 2. S = {u,u} .

t

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.

52)

(w) = ( , ) = (-4/2 , 10/2) = (-22 ,, [w] = [-22 52]

1.ii) (w) = (a,b) w = au + bu : (1,1) = a(2,-4) + b(3,8) : 2a + 3b = 1 a = 5/28 -4a + 8b = 1 b = 3/14 , [w] = 5/28

3/14

4.i) u = cv + cv + cv

:

-3 -6c + 2c - 2c 0 = -6c - 6c - 3c -3 4c + 7c

: c = 3/4, c = -3/4, c = 0. u u : B' = {v,v,v} '.

=

3/4 3/4 1/12 -3/4 -17/12 -17/12 0 2/3 2/3

ii) a, b, c w = au + bu + cu : -3a -3b+c = -5 a=1 2b+6c= 8 b=1 -3a -b- c = -5 c=1 [w] = [1 1 1] t

}

1 [w]' = P[w] = P 1 1

19/12 = -43/12 . 4/3

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..O 6: .{e1,e2,,en} . :Rn Rm

:Rn R

Rn. (), mxn

11: x y z x +y +z -7x +z 0 3y

.

:R3 R4

=

. ()

:

1 0 0

=

1 +0 +0 -7*1 +0 +0 0 3*0

=

1 -7 0 0

0 1 0

=

1 0 0 3 1 -7 0 0 1 0 0 3

0 0 1

=

1 1 0 0

()=

1 1 0 0

103 24 52

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4: .

()

x Rn

:Rn Rm.

T(x)=M(T)x

12:6 -5 6

. .

. 11.

(.4)6 -5 6 = 1 -7 0 0 1 0 0 3 1 1 0 0 6 -5 6 = 7 -36 0 -15

13: . -4 6 (e1)= ,T(e2)= 0 2 ,T(e3)= 1 -2

:R4 R23 3 ,T(e4)=

4 -4 5 -4

.

()=-4 0 1 6 2 -2 3 3

(.4)

4 -4 5 -4

=

-4 0 1 6 2 -2

3 3

4 -4 5 -4

=

-23 -6

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104

5:. .

mxn

:Rn Rm

(x)=Ax. .

()=.

: ()

5

x Rn.

(x)=M(T)x ,

6::U V

. .

S:V W. SoT:U V ..

7::Rn Rm

. .

S:Rm Rk. .. SoT:Rn Rk

M(SoT)=M(S) M(T)

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105

14:x y z

3x + 6z -5x +2y +2z 3x +y -4z

:R

3

R

3

S:R

3

R

4

.

=

x y z

3x + 3y +3z x - y +z -2x + y 10y + 3z =

,

S

N S.T

x y z x y z 3x + 6z -5x +2y +2z 3x +y -4z

ST S

o

=

S(T

)=

=

3(3x + 6z) +3(-5x + 2y + 2z) +3(3x + y 4z) 3x + 6z -(-5x + 2y + 2z) + 3x + y 4z -2(3x + 6z) + (-5x + 2y + 2z) 10(-5x + 2y + 2z) + 3(3x + y 4z)

=

3x + 9y +12z 11 x - y 11x + 2y - 10z -41x +23y + 8z

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15:

. 14

106 .7.

T(x)=

3 0 6 -5 2 2 3 1 -4

S(x)=

3 3 1 -1 -2 1 0 10

3 1 0 3

T

(.7) (SoT) = M(S) M(T)

=

3 3 1 -1 -2 1 0 10

3 1 0 3

3 0 6 -5 2 2 3 1 -4

=

3 11 -11 -41

9 -1 2 23

12 0 -10 8

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107 To rauK . . rauK

1

.

8: :

.

:Rn Rm. ()

i) H ii)

rauK T = rauK M(T) . . :R3 R4

16:

(e1) = (9,8,9,8) , T(e2) = (-1,-1,-1,0) , T(e3) = (-3,5,4,-6) N (1,1,1).

()=

9 -1 -3 8 -1 5 8 -1 4 8 0 -6

|1,1,1| = M(T)

t

1 1 1

=

5 12 12 2

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(1,1,1) = (5,12,12,2) 108

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. , GRAM SCHMIDT

.

: V . . . V . 1 . : S = {1,2,3} 1 = ( 0, 1, 0 ), 2 = ( 1/, 0, 1/ ), 3 = (1/, 0, -1/ ) 3 . . : < 1, 2 > = < 1, 3 > = < 2, 3 > = 0 1 = 2 = 3 = 1 : V . . 0V . ___ 1. ___ / 0 . . : S = {u1, u2, u3} u1 = ( 0, 1, 0 ), u2 = ( 1, 0, 1 ), u3 = ( 1, 0, -1 ) S . S = {u1, u2, u3 }, u2 = _____ . : V . . S = { 1, 2,...,n } V uV, u = < u, 1 > 1 + < u, 2 > 2 +...+ < u, n > n. : u = 11 + 22 +...+ nn < u, i > = < 11 + 22 ++ nn, i > = 1 < 1, i > + 2 < 2,i > ++ n < n, n > (1) 2 S < i, i > = ||i|| = 1 < i, j > = 0 , ij (1) < u, i > = i : S = {1, 2, 3 } 1 = ( 0, 1, 0 ) , 2 = ( -__, 0, __ ) , 3 = ( __, 0, __ ) u = (1, 1, 1) . 1, 2, 3 : V . . S = { 1,2,,n } V. S . . : 11 + 22 ++ nn = 0 (1) i S < 11 + 22 ++ nn, i > = < 0, i > = 0 1 < 1, i > + 2 < 2,i > ++ n < n, n > = 0 (2) S . < yj, yi >, ij < i, i > = 1 (3) (2), (3) < i, i > = 0 i = 0 S . : V . . {1, 2,, r } . W 1, 2,, r u V U = w1 + w2 w1 W w2 W w1 = < u, v1 > v1 + < u, v2 > v2 ++ < u, vr > vr (1) w2 = < u, v1 > v1 - < u, v2 > v2 -- < u, vr > vr (2) :

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w1 u W projwu w2 = u projwu u W. : 3 . . . W v1 = ( 0, 1, 0 ) v2 = ( . u = ( 1, 1, 1 ) : projwu = < u, v1 > v1 + < u, v2 > v2 =(1) ( 0, 1, 0 ) + ( __ ) ( -__, 0, __ ) = ( __, 1, __ ) . u W u projwu = ( 1, 1, 1 ) ( __, 1, __ ) = ( __, 0, __ ) u projwu 1, 2 , W 1,2. : . . . : V S = { u1, u2, , un } V . { v1, v2, , vn } V. 1 1 = ____. 1 1. 2 2 1 1, u2 W, 1 . : 2 = ______ u2 - < u2, v1 > v1 = 0 u2 = < u2, v1 > v1 = < u2, v1 > __ = ____ u1 u2 / u1. S = { u1, u2, , un } . 3 _____________________ 4 _________________ .. Gram Schmidt. : 3 . . . G S u1 = ( 1, 1, 1 ), u2 = ( 0, 1, 1 ), u3 = (0, 0, 1 ) . : 1 / 1 = _________ 2 / u2 p...