Unidad 3 Espacios Vectoriales
February 15, 2023 | Author: Anonymous | Category: N/A
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ZMFENE 0 JP_N@FGP XJ@UGQFNDJP
@NQDGP BNMZJD PNQBFJMUG Uutgr
KZDFN @NQGDFMN CZJQQJQG _DN[NP @`>??103?183
CQZ_G 0>8
ZMFXJQ@FENE MN@FGMND NAFJQUN V N EFPUNM@FN (ZMNE) NDCJAQN NDCJAQ N DFMJND DFMJND >1/>>/>9
JKJQ@F@FGP Jkjr`f`fg >; @. FMEJ_MEJM@FN FMEJ_MEJM@FN J FMEJ_JMEJM@FN FMEJ_JMEJM@FN DFMJND
Jkjr`f`fg 1; 1 Enegs; n) Y 2 4 >,0,: >,0,: = < V 2 4 1,5,: 1,5,:=< =< [24 [24 >,?,1 >,?,1=< =< vj`t vj`tgr grjs js quj quj pjrtj pjrtjmj mj`j `jm m n um jspn jspn`f `fg g vj vj`t `tgr grfn fndd X, ejbu ejbujs jstr trj j jd nxfg nxfgbn bn mûbj mûbjrg rg 1 ejmg ejmgbf bfmn mneg eg Djy Djy `gmbutntfvn ej dn subn ej vj`tgrjs. u + v 2 v +u
⃓
⃓
⃓
⃓
x + y + z 2( >,1,0 ) + ( 1,5,: ) +( >,?,1)
⃓
⃓
⃓
x + y + z 2( 5,8,>? )
⃓
⃓
⃓
a) Pfjmeg Pfjmeg ξ y α vnrfnadjs vnrfnadjs js`ndnr js`ndnrjs, js, ejbujstr ejbujstrj j jd sæptfbg sæptfbg y g`tnvg nxfgbn nxfgbn pnrn jspn`fgs vj`tgrfndjs usnmeg dgs vj`tgrjs ejd jspn`fg vj`tgrfnd X ejd pumtg nmtjrfgr. Zsj vndgrjs ej 0 y 5 pnrn ξ y α rjspj`tfvnbjmtj. ξ(Y + V + [) 2 ξ Y + ξ V+ ξ [ (_rfbjrn djy efstrfautfvn) (ξ + α)Y 2 ξ Y + α Y (Pjcumen djy efstrfautfvn) ξ ( Y + V + [ ) 2ξ Y + ξ V + ξ [ 0 (( >,1,0 )+( 1,5,: )+( >,?,1 ))2 0 ( >,1,0 )+ 0 ( 1,5,: )+ 0 ( >,?,1 )
(0,8,6 )+(8,>1 , >: )+(0,?,8 )2(0,8,6)+ )+((8,>1,>: )+( 0,?,8 )
( >1,>9,0? )2( >1,>9,0? )
(ξ + α ) Y 2ξ Y + α Y (0 + 5 )( >,1,0 )20 ( >,1,0 )+ 5 ( >,1,0 ) 3 ( >,1,0 )2 0 ( >,1,0 )+ 5 ( >,1,0)
(3,>5,1> )2 )2((0,8,6 )+( 5,9,>1 ) (3,>5,1> )2 )2((3,>5 , 1> ) Ejs`rfp`fðm ejd jkjr`f`fg 1 n) Eneg neg jd `g `gmkum mkumtg tg s2 u> , u1 egmej u>2( :,> ) y u12(∑0 , ∑1) . Ejbgstrnr quj P cjmjrn n Q1 s2 { ( :,> ) , (∑0 , ∑1 ) }
( x , y ) ∉ Q1 ( x , y ) 2ξ ( :,> ) + α (∑0 , ∑1 ) ( x , y ) 2( : ξ , ξ ) + (∑0 α ,∑1 α ) ( x , y ) 2( : ξ ∑ 0 α , ξ ∑1 α )
: ξ ∑0 α 2 x ,ξ ∑1 α 2 y : ξ ∑0 α 2 x
{
ξ ∑1 α 2 y
Pgdu`fgmnbgs jd sfstjbn pgr jd bætgeg ej rjeu``fðm cnussfnmn : >
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: ∑0 x ∑0 x h 1 1 2·->+h1 ∑1 y ? ∑0 ∑ y
∑0 α 2∑ y α 2∑ y + 0 : ξ ∑0 α 2 x : ξ ∑0 (∑ y + 0 )2 x : ξ + 0 y ∑6 2 x : ξ 2 x ∑ 0 y + 6
ξ 2
x ∑0 y + 6 :
Jmtgm`js ej`fbgs quj jd `gmkumtg s `gmhgrbn um sfstjbn cjmjrnegr ej r1.
( x , y ) 2ξ ( :,> ) + α (∑0 , ∑1 ) ( x , y ) 2 x ∑0 y + 6 ∓( :,> )∑ y + 0∓(∑0 ,∑1 ) :
`grrj`tg nhfrbnr quj quj jd vj`tgr a) Enegs dgs vj`tgrjs u 2∑8 f+ 6 k y v 2∑f + 6 k ·js `grrj`tg w 2∑>> f∑ 6 k js umn `gbafmn`fðm dfmjnd ej u y v7 Kustfhf`nr dn rjspujstn. ⃓
⃓
Ujmjbgs quj ejbgstrnr quj w 2l > u + l 1 v
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⟩ ∑>> ,∑6 ⟦2 l > ⟩∑8,6 ⟦ + l 1 ⟩∑>,6 ⟦
⟩ ∑>> ,∑6 ⟦2 ⟩∑ 8 l > , 6 l > ⟦ + ⟩∑l 1 , 6 l 1 ⟦ ⟩ ∑>> ,∑6 ⟦2 ⟩∑ 8 l >∑l 1 , 6 l > + 6 l 1 ⟦ In`jbgs fcundene ej vj`tgrjs
⃓
∑8 l >∑l 12∑>> 6 l >+ 6 l 12∑6
Gatjmjbgs jd sfcufjmtj sfstjbn ej j`un`fgmjs y rjsgdvjbgs pgr jd bætgeg ej rjeu``fðm cnussfnmn
∑8 l >∑ l 12∑>> 6 l >+ 6 l 12∑6
{
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∑8 ∑> ∑>> 0 h > ∑8 ∑> ∑>> > + 1 h 1 1 2h 1 1 6 6 ∑6 ? >: ∑:>
>: l 12∑:>
l 12 l 12
∑:> >:
∑>3 :
∑8 l >∑l 12∑>> ∑8 l >∑( ∑8 l >+
∑>3 :
>3 :
)2∑ )2 ∑>>
2∑>>
∑8 l >2∑>>∑
>3 :
∑8 l >2∑:31 ∑>
(∑8 l )2 8 >
l >2
31
l >2
>1
0?
:
Pgdu`fðm
∑> ∑31 ( ) 8
:
l 12
w2
⃓
∑>3 :
∑>3 :
l >2
u+
⃓
>1 :
>1 :
v
⃓
Pj ejbgstrð quj jd vj`tgr w js `gbafmn`fðm dfmjnd ej dgs vj`tgrjs u y v yn quj jxfstjm dgs js`ndnrjs y sgm mûbjrgs rjndjs. Ejs`rfp`fðm ejd jkjr`f`fg 5 Ej dn sfcufjmtj bntrfz quj ncrupn n trjs vj`tgrjs ej um jspn`fg vj`tgrfnd, `nd`udj;
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3 0 5
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∑6 ∑>> 5 ∑> ∑>0 ∑>?
n) Ejtj Ejtjrb rbfm fmnm nmtj tj
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3 0 5
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∑6 ∑>> 5 ∑> ∑>0 ∑>?
Ztfdfznbgs dn djy ej snrrus
∑6 ∑>> 0 5 ∑> |5 ∑>0 ∑>? | 3 ∑6 ∑>> 0 5 ∑> 3
N 2 (-19?+516+08 (-19?+516+08)) - (->38+6>+13?) (->38+6>+13?) N 2 >9: - >9: N 2 ? Dgs vj`tgrjs sgm sgm dfmjndbjmtj ejpjmefjmtjs
a) Qnmcg Ztfdfznbgs jd bætgeg ej rjeu``fðm cnussfnmn 3 0 5
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3 3 ∑6 ∑>> 3 ∑6 ∑>> ∑6 ∑>> :: 18 H 1 + H 0 2 H 0 ? :: 18 ∑ 5 H > +3 H 0 2 H 0 ? 5 ∑> ∑0 H > + 3 H 1 2 H 1 ? ? ? ∑:: ∑18 5 ∑>0 ∑>? ∑>0 ∑>?
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Q21 Dn bntrfz js dfmjndbjmtj fmejpjmefjmtj
`) Bntrfz Bntrfz js`ndgm js`ndgmnen nen usnm usnmeg eg Cnuss Cnuss Kgreî Kgreîm m In`jbgs dn `gbafmn`fðm dfmjnd n ( 3,0,5 )+ a (∑6,5 , ∑>0 ) + ` (∑>> ,∑> ,∑>? )2( ?,?,?)
( 3 n , 0 n , 5 n ) + (∑6 a , 5 a ,∑>0 a )+ (∑>> ` , ∑> ` , ∑>? ` ) 2( ?,?,? )
{
3 n∑6 a ∑>> ` 2 ? 0 n + 5 a∑ ` 2? 5 n∑>0 a ∑>? ` 2?
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3 0 5
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∑6 ∑>> ? > + 3 h 1 1 2h 1 1 5 ∑> ? ∑0 h > ∑>0 ∑>? ?
`2 ? :: a + 18 ` 2? :: a + 18 ( ? )2? :: a 2? ?
a 2 :: a 2? 3 n ∑6 a ∑>> ` 2? 3 n ∑6 ( ?)∑ >>( ? )2 ? 3 n 2?
n2
? 3
n 2?
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3 ? 5
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∑ 6 ∑>> ?
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3 :: 18 ? ∑5 H > + 3 H 02 H 0 ? ? ∑ >0 ∑>? ?
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∑ 6 ∑>> ?
:: ∑::
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3 1+ H 0 02 H 0 ? 18 ? H 1 ? ∑18 ?
Pgdu`fðm n 2? a 2? ` 2?
Dgs vj`tgrjs sgm dfmjndbjmtj fmejpjmefjmtjs e) Kustfhfquj Kustfhfquj ejsej `nen `nen prg`jsg prg`jsg sf iny ejpjmejm` ejpjmejm`fn fn g fmejpjmejm`fn fmejpjmejm`fn dfmjnd. dfmjnd. Ejs`rfp`fðm ejd jkjr`f`fg : Ejtjrbfmj fmejpjmejm`fn dfmjnd ej dgs sfcufjmtjs `gmkumtgs ej vj`tgrjs. n. X>2 (?, (?, 1,1). 1,1). X12 X12 (0,0,0 (0,0,0). ). X02 X02 (?,?,5) (?,?,5).. Npdf`nrjbgs Npdf`nrj bgs dn djy ej snrrus snrrus pnrn ejtjrbfmnr ejtjrbfmnr su ejtjrbfmnmtj ejtjrbfmnmtj ? 0
?
1 0
?
|1
5|
0
? 0
?
1 0
?
N2 (?+?+?)-(?+?+15 (?+?+?)-(?+?+15)) N2-15 Ej`fbgs quj dgs vj`tgrjs sgm dfmjndbjmtj fmejpjmefjmtjs a. X>2 (8 ,-1, 9 ). X12 (>/1, 5, 5, ?) . X02 (->?, (->?, 8, 1). 1). X52(1,>,5). X52(1,>,5). Inddnbgs jd rnmcg ej dn bntrfz pgr jd bætgeg ej grdnr
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|∑ / |2 8 1
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8
∑1 9
8
∑1
> 1 5
> 1 5 ? > 1 5 ?
15 ∑( ∑> ) 21: ≯ ?
8 1
2( 59 + 15 + ? )∑(∑01? ∑1 + ? )231 + 011 2065 ≯ ?
1 > 5
2( 68 + 5 + ? )∑( 85 ∑ 5 + ? )2>??∑ 8?2 5? ≯ ?
| | 9
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∑>?
Ej`fbgs quj jd rnmcg ej dn bntrfz js Q20 pgr `und dgs vj`tgrjs sgm dfmjndbjmtj fmejpjmefjmtjs Ejs`rfp`fðm ejd jkjr`f`fg 8 Ejbgstrnr dg sfcufjmtj; Pf N y A sgm sgm bntrf` bntrf`js, js, ejbuj ejbujstr strj j dns dns sfcuf sfcufjm jmtj tjs s prgp prgpfje fjenej nejs s y `gbp `gbprga rganr nr bjefnmtj jkjbpdg; n. Qnmcg (NA)2 rnmcg rnmcg (At Nt ) tjmcn prjsjmtj jd grejm ej dns bntrf`js. bntrf`js.
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N 2
> :
0 A 8
1 5
? >
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Qnmcg (
Qnmcg
> :
0 1 ∓ 8 5
| | >5 05
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? )2rnmcg ( 1 > ?
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0 2rnmcg >5 8 0
5 > ∓ > 0
: 8
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05 8
rnmcg ( 95 ∑>?1 ) 2rnmcg ( 95 ∑>?1 ) rnmcg (∑>9 ≯ ?)2 rnmcg (∑ >9 ≯ ? )
rnmcg 1 2rnmcg 1
a. Pf N mg js umn bntrfz `unernen, dgs vj`tgrjs hfdn g dgs vj`tgrjs ej `gdubmn ej N sjrîm dfmjndbjmtj dfmjndbjmtj ejpjmefjmtjs. ejpjmefjmtjs.
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