<NOTE>

<HEAD1>@{<E>Tensor product</E>@}</HEAD1>

AmrA ekhAne @{<E>tensor product</E>@} nAme ekTA jinis shikhba. er sa.njnATA ekTu adbhutbhAbe dite hay. eirakam adbhut

sa.njnAr ekTA kAydA Achhe, seTA Age nA jAnA thAkle @{<E>tensor product</E>@} bojhA shak+ta. tAi sei byApArTA Age alochanA

kare neba.

<HEAD2>ghuriye nAk dekhAno</HEAD2>

dharo ballAm @{<M>f:\rr\to\rr</M>@} hala ekTA @{<E>linear function.</E>@} tAhale balte pAro @{<M>f(x)</M>@} kI rakam

dekhte habe? uttarTA nishchayai jAno, @{<M>f(x) = ax</M>@} jAtIya kichhu ekTA. eiTA hala sojAbhAbe balA. ghuriye balAr

kAydA hala, eiTA balA ye @{<M>f(x) = x</M>@} hala sab @{<E>linear function</E>@}-er nATer guru. bAkIrA @{<E>linear function</E>@}-rA

sakalei er @{<E>multiple.</E>@} eibhAbe ``nATer guru'' eba.n bAkIderke ``tAr sange sampar+ka'' diye prakAsh karA. ArekTA udAharaN

dekhi. ebAr ballAm @{<M>f:\rr^2\to\rr^3</M>@} hala ekTA @{<E>linear function.</E>@} er chehArA kIrakam? ekhAneo sojAsujibhAbe

uttar habe @{<M>f(x,y)=(a_{11} x + a_{12}y,\, a_{21}x+a_{22}y,\, a_{31}x+a_{32}y),</M>@} yekhAne @{<M>a_{ij}</M>@}-gulo yA khushi

kichhu sa.nkhyA. ekhAne nATer guru ekjan nay, chhayjan--

@{<MULTILINE>

f_1(x,y) & = & (x,0,0),\\

f_2(x,y) & = & (0,x,0),\\

f_3(x,y) & = & (0,0,x),\\

f_4(x,y) & = & (y,0,0),\\

f_5(x,y) & = & (0,y,0),\\

f_6(x,y) & = & (0,0,y).

</MULTILINE>@}

erA sakalei ekekTA @{<E>linear function</E>.@} shudhu tAi nay, anya yekono @{<E>linear function</E>@}-kei ei kaTAr @{<E>linear combination</E>@}

hisebe lekhA yAy. ebAr mane karo ei nATer gurur tAlikAy Arekjanke juRe dilAm--

@{<D>f_7(x,y) = (x,y,0).</D>@}

balAi bAhulya yekono @{<E>linear function</E>@}-kei ei sAtjaner @{<E>linear combination</E>@} hisebeo lekhA yAbe (khAli

@{<M>f_7</M>@}-er @{<E>coefficient</E>@}-TA shUnya nilei hala). kintu laxa karo eTAi ekmAtra @{<E>linear combination</E>@}

nay, kAraN @{<M>f_1 + f_6 = f_7</M>@} hachchhe, tAi @{<M>f_1</M>@}-er badale @{<M>f_7-f_6</M>@} likhle yekono @{<E>linear combination</E>@}

theke ArekTA natun @{<E>linear combination</E>@} pAoyA yAbe. sutarA.n laxa karo, nATer gurur tAlikAy anAbashyak sadasya

yog halei Ter pAoyA yAbe @{<E>linear combination</E>@}-er @{<E>nonunique</E>@} haoyA theke.

<P/>

sutarA.n nAk ghuriye balAr kAydATA d,nARAchchhe eirakam--eman kichhu nATer gurur tAlikA deoyA bAkIderke

yAder @{<E>unique</E>@}

@{<E>linear combination</E>@} hisebe lekhA yAbe. eibAr ArekTu kaThin udAharaNer samay esechhe.

<P/>

ebAr AmrA eman sab @{<M>f:\rr^2\to\rr</M>@} pete chAi yArA @{<E>bilinear.</E>@} arthA.t @{<M>x</M>@} sthir thAkle @{<M>y\mapsto f(x,y)</M>@}

habe @{<E>linear function,</E>@} AbAr @{<M>y </M>@} sthir thAkle @{<M>x\mapsto f(x,y)</M>@}-o habe ekTA @{<E>linear function.</E>@}

ekhAneo sojA uttar hala @{<M>f(x,y) = axy.</M>@} sutarA.n nATer guru ekTAi, @{<M>f_1(x,y) = xy.</M>@} yadi @{<M>f:\rr^2\times\rr^3\to\rr</M>@}-ke

@{<E>bilinear</E>@} chAitAm tabe ekai yuk+tite sojAsuji uttar hala

@{<D>f (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) = a_{11}x_1y_1+a_{12}x_1y_2+a_{13}x_1y_3+a_{21}x_2y_1+a_{22}x_2y_2+a_{23}x_2y_3,</D>@}

yekhAne @{<M>a_{ij}</M>@}-rA yA khushi. nAk ghuriye balle

nATer guru neoyA yAy ei chhayjan-ke-

@{<MULTILINE>

f_1 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_1 y_1\\

f_2 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_1 y_2\\

f_3 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_1 y_3\\

f_4 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_2 y_1\\

f_5 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_2 y_2\\

f_6 (#( (x_1,x_2),\, (y_1,y_2,y_3) )#) & = & x_2 y_3.

</MULTILINE>@}

ei tAlikAy kono bAhulya nei, kAraN anya yekono @{<E>bilinear</E>@} @{<E>function</E>@}-ke eder @{<E>unique</E>@} @{<E>linear combination</E>@}

hisebe lekhA yAy.

<HEAD2>sojAsuji sa.njnAr samasyA</HEAD2>

AmrA ei kAydAy yekono @{<E>bilinear</E>@} @{<E>function</E>@}

@{<M>f:\rr^m\times\rr^n\to\rr</M>@} -ke sojAsuji likhe felte pAri. kintu byApArTAke ArekTu @{<E>generalise</E>@} karlei

sojAsuji sa.njnA deoyA asambhab haye yAy, takhan oi ghuriye nAk dekhAno chhARA path thAke nA. yeman yadi @{<M>\rr^m</M>@}-er

jAygAy kono @{<E>infinite dimensional vector space</E>@} nii, bA kono @{<E>module</E>@} nii. sekhAne ghuriye nAk dekhAnor

kaydATA karlei AmrA @{<E>tensor product</E>@}-e ese upasthit haba. kIbhAbe bali.

<P/>

AmrA duTo @{<E>module</E>@} nichchhi @{<M>M</M>@} Ar @{<M>N.</M>@} ebAr AmrA @{<M>M\times

N</M>@}-er upare @{<E>defined</E>@} yata rakamer @{<E>bilinear</E>@}

@{<E>function</E>@} hay, tAder chehArA bojhAr cheSTA karchhi. tAr janya eder madhye nATer

guru kh,nuje pete habe, mAne eman

kichhu @{<E>bilinear</E>@} @{<E>function</E>@},

yAder byabahAr kare bAkIder @{<E>linear</E>@}-bhAbe likhe felA yAbe @{<E>unique</E>@}-bhAbe. etxaN to AgebhAge sojA kAydA

byabahAr kare AndAj kare felchhilAm kAder nATer guru nile kAj habe. kintu ekhan to sojA kAydA bandha, ataeb AmrAo andha.

satyi balte ki eman nATer guru Ad\ou pAoyA yAbe kinA tArao khub ekTA nishchayatA chokhe paRchhe nA. sukher kathA ye, pAoyA

yAbe. ei AshwAs~bANITAke ekTA @{<E>theorem</E>@}-er AkAre lekhAr cheSTA karA yAk, pramANer kathA tAr pare. nATer guru swaya.n

ekjan @{<E>bilinear</E>@} @{<E>function</E>.@} er @{<E>domain</E>@}-TA to @{<M>M\times N,</M>@} kintu @{<E>codomain</E>@}-TA

kI? ke jAne, kichhu ekTA! tAr ekTA nAm deoyA yAk, @{<M>T.</M>@} Ar nATer guru kinA @{<E><RED>g</RED>uru</E>@}, tAi t,nAr

nAm dilAm @{<M>g.</M>@} arthA.t @{<M>g:M\times N\to T</M>@} halen AmAder upAsya sei gurudeb. gurur dui mahAguN--

<UL><LI>ek, uni nije @{<E>bilinear</E>@},</LI>

<LI>dui, anya yekono @{<E>bilinear</E>@} @{<E>function</E>@} gurudeber shiSya (mAne @{<E>linear

function</E>@}), arthA.t yadi kono @{<E>bilinear</E>@} @{<E>function</E>@} nAo @{<M>f:M\times

N\to P</M>@} (yA khushi @{<M>P</M>@} hate pAre), amni dekhbe @{<M>f(x)</M>@}-ke @{<M>f(x) =

f'(g(x))</M>@} AkAre lekhA yAy, yekhAne @{<M>f'</M>@} kono @{<E>linear function.</E>@}</LI>

<LI>tin, ei @{<M>f'</M>@}-TA AbAr @{<E>unique</E>@}-o baTe, arthA.t AmAder guru med~bAhulyabar+jita!</LI>

</UL>

eman guru ekTA antata.H sar+badAi pAoyA yAbe, mAne yekon @{<M>M</M>@} Ar @{<M>N</M>@}-er janyai. ekai @{<M>M, N</M>@}-er

janya ekAdhik erakam guru pAoyA yAy nA? xepechho? ekTA ye pAoyA yAchchhe, sei AmAder chod+dopuruSer bhAgyi, AbAr Aro chAy!

nA, yekono @{<M>M,N</M>@}-er janya erakam guru Thik ek pIsai pAoyA yAy. abashya gurudeb nAnArUpe lIlA karte pAren. sei rUpabhed dekhe

yena AlAdA guru mane kare boso nA. hAjAr hok, jAnoi to yini Al+lA tinii mA shItalA. erakam ekekTA rUpdhAraNke bale @{<E>isomorphism.</E>@}

tAi balba ye gurudeb hale @{<E>unique</E>@} @{<E>upto isomorphism.</E>@}

<P/>

eTAi hala AmAder @{<E>theorem.</E>@}

<P/>

gurudeb AmAder baRa Adarer, tAi bhak+terA t,nAke bhAlobese DAknAm diyechhen. bishwabramhAnDAnandas+wAmIr DAknAm yeman pal+TumahArAj

hate pAre, seirakam Ar ki! oi ye @{<M>T</M>@}-er kathA ballAm oTAke bale @{<M>M</M>@} Ar

@{<M>N</M>@}-er @{<E>tensor product</E>@} eba.n lekhe @{<M>M\otimes N.</M>@} oi @{<M>g</M>@}-TAkeo loke @{<E>tensor product</E>@}

bale, Ar @{<M>m\in M</M>@} Ar @{<M>n\in N</M>@} hale @{<M>g(m,n)</M>@}-ke lekhe @{<M>m\otimes n.</M>@}

<P/>

ei ab.hdhi shune ekTA khTkA b,nAdhte pAre. AmAder AgekAr ekTA udAharaNe to chhayjan nATer guru chhila, tabe ekhAne kI kare

ekjanei kAj haye gela? kAraN oi chhayjanke pyAk kare AmrA ekTA pyAkeT bAniyechhi. yeman sekhAne @{<M>f_1,...,f_6</M>@}

chhila. orA sabAi chhila @{<M>\rr^2\times \rr^3\to\rr.</M>@} AmrA oderke parpar basiye @{<M>g = (f_1,...,f_6)</M>@} karechhi,

yekhAne @{<M>g:\rr^2\times \rr^3\to\rr^6.</M>@} byas.h!

<P/>

yAi hok gurudeber barNanA anek hala. gurudeber as+tit+wer pramAN AbAr pare karba.

<DISQUSB id="tensor"

</NOTE>