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wªl«|¬pK«pmcc/tensorbook/ch0.pdf · u,xxxn^ixnz#h giqnu,civ,hqnf0qnriu,xxgiv,f&b 'y h#lnhyp+lnh xnf+m0hlnh >girilnp+xnh#em`+h lnxnu,f+dixlf+j qnrihy\!^ih#xnqnu*f+dtsslnfw^ jwr

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Page 1: wªl«|¬pK«pmcc/tensorbook/ch0.pdf · u,xxxn^ixnz#h giqnu,civ,hqnf0qnriu,xxgiv,f&b 'y h#lnhyp+lnh xnf+m0hlnh >girilnp+xnh#em`+h lnxnu,f+dixlf+j qnrihy\!^ih#xnqnu*f+dtsslnfw^ jwr

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Ù �-� º Û ¹ ¼ � Ó�Ù �-� º Û ¹ ¼ ���-� º Óñ¼ ���-� º ÞCEDgIP+LNQNU*Z#^IV*P+L&S|QNRIHKJfPWM0U,V*U,P+LYV,H�P+XNQN�>XN\!^IP+LNH�XigILNF&¼NH�Z#QNU*F+DMP+QNLNU,Z�H#XY´ � Ó�°�Ù�°<¹]¼½°�ÛKº Ï °�¹�¼ P+DIe ¶:Ú ´ �SP+LNHXNH�V,Jf�>Z�F+D&¼N^Ij+PWQNH0PWD�ehXNH#V*Jf�>U*D!`WH#LNXNH0U,DhQNR�Hvn~F!F+LNH�� � H�DILNF+XNH0XNH�D�XNH 'q� RIH#XNHM0PWQNLNU,Z�H#XyP+LNHZ#FWD�XNU*eIH#LNH�e�P+XQNLNP+DIXNJfFWLNM0P+QNU,FWD�XlJfLNF+M½QNRIH�XNPWM0gIV,H�XNgIP+Z�H|,ØQNF0U*QNXNH#V*JNS!QNRIHyU,DIDIH#LlgILNF!eI^�Z�QlM0P+QNLNU*��c(H#U*DIj ¼ '¤ U*£(H#LNH�DaQ�Z=RIF+U,Z�H�X�F+JyU*D�DIH�L�g�LNFae�^IZ�Q�M0P+QNLNU,Z�H�X�FWD�QNR�HrQEk:FnXNgIP+Z#H�X�b!U*H#V*e�eIU*£wH�LNH�D!QXn~F!F+LNH�� � H�DILNF+XNHU,Da`+H�LNXNH�X&S(P�g(F+U*D!Q]QNRIP+Q]U,XzLNP+LNH�V,bmH�M0gIRIP+XNU,��H�e ' CED�Z#FWM0M0F+D�klU,QNR~M0F+XNQzP+^IQNRIF+LNU*QNU,H�X&S � P+F Ù )�Ü � � S�g ' �W� ÛP+DIe�çyLN^�XNd&P+V Ù )�Ü � � Û QNP+d+H ¼ �PWD�e ¼ �yQNFc(H]QNR�HzU*e�H�DaQNU,QEb�M0P+QNLNU,Z�H�X 'lÿ QNF+DIH � XKe�H��IDIU,QNU,FWD�Z#F+U*DIZ#U*e�H�XKklU,QNRF+^ILNX 'dSe�6f-47JL9�u��[i�d�01J;Z5MlD3E�f�ML6J�8E¨Hâ6�IKMl6�D389|f�6[Z5IKMle%m_ö�H#QTÍ�c(HyP+DmU,DIDIH�LxgILNF!eI^IZ#QlXNgIP+Z�H�klU,QNR}M0H�QNLNU,ZyQNH#DIXNF+L¼ Ó ë=ý �L� ï S_XNb!M0M0H�QNLNU,Z�P+DIeng(F+XNU*QNU,`+H�eIH���DIU*QNH!S_P+DIe¡V,H�Q|, Ó Í ã cwH0QNRIH�eI^�PWVvF+J:Í '0� RIH�eI^�PWVvU,DIDIH#LgILNF!eI^IZ#QmMP+QNLNU,�3U,D�Í�ã�U,X ¼Ïº Ï ' CED�XNQNPWQNU,XNQNU,Z�P+V�P+gIgIV,U*Z#PWQNU,F+DIX&S ¼Ïº Ï Ó¢¡ U,Xr^IXN^IP+V*V,b3P{Z#F&`&P+LNU*P+DIZ#HM0P+QNLNU*� 'n� Z#FWD!QNLNP5`|P+LNU*P+DaQ�QNH�DIXNF+L0klU*QNR�Z#F+MgwFWD�H�DaQNX Õ �L� M0P&b�c(H}LNH#jWP+LNeIH#e�P+X0QNRIH}Z#F+MgwFWD�H�DaQNX�FWJ]PV,U*DIH#P+LyQNLNP+DIXNJfFWLNM�JfLNF+M¸, Ó Í�ãzQNF�Í 'q� Ra^IX ¼ � Ów¡ PWD�e ¼ � Ów¡Pº Ï '}ÿ ^IgIgwFWXNH�QNRIP+Q��ØU,X�jWU,`+H�D�U*DM0P+QNLNU*��DIF+QNP+QNU*F+D}c!b

� Óñ¼ � Ù�¶Ú�°�Ù�° ¹ ¼ � °�Û º Ï ° ¹ ¼ � Û(Óñ¼ � Ù�¶Ú ´ Û ÖklRIH#LNH ° U*XKPGj+U*`+H#D�M0P+QNLNU,�0klRIF+XNHzZ�F+V*^�MD�XTP+LNH]V,U*D�H�P+LNV*b�U*D�eIH�g(H#DIeIH#DaQT`+H�Z#QNFWLNXTU,Db, '�� RIH�D�QNRIH-n~F!F+LNH��� H�DILNF+XNH�U,Da`+H�LNXNH�U,X

� º Ó Ù�¶Ú ´ Û`¼ �CED}gIP+LNQNU,Z�^IV,P+L&S!U,J ¼ � Ó eIU,P+j ë � Ï´Ö Þ�Þ#Þ Ö � �wï klU*QNR � � Ó ) SaP+DIe °ÝÓ ) S!QNRIHyZ�F+DIXNQNP+DaQx`+H�Z�QNF+L&SaQNRIH#D���U,XQNRIH�Mq^�V*QNU,DIF+M0U*P+V�Z�F&`&P+LNU,P+DIZ�H�MP+QNLNU,��klRIF+XNH�n~F!FWLNH#� � H�DILNF+XNH�U,Da`+H�LNXNH�U,X

e�U*P+j ë ) � � Ï�Ö Þ�Þ#Þ Ö ) � � �(ïlÚ )+) ¹ ÞCED}QNRIU,XlU,DIXNQNP+DIZ�H!S!QNRIHyDa^IV,V_XNgIP+Z�H�F+J+��U*XxQNRIHyXNgIP+DmFWJ ) SaklR�U*V,HyQNRIHyDa^�V*V_XNg�PWZ#H�F+J+� º U*XlQNR�HyXNgIP+D}F+J�QNRIH`+H�Z#QNFWL � '

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s: Ò Ó�É Ò � s: � PWLNH�QNR�HyZ�F+M0g(F+DIH#DaQNXlF+J s:mÕ Í Ï klU,QNR}LNH�XNg(H#Z�QlQNF0QNRIH�cIP+XNU*X ë� Ò ï ' CEDmXNb!MqcwFWV,X&S

s:Ó s: Ò � Ò Ó�É Ò � � Òè � � Ö�:%éU,XlQNRIH�gILNF&¼NH�Z#QNU*F+D}F+J : F+DmQNF[Í Ï '� F[klLNU,QNH�QNRIH]XNP+M0H]QNR�U*DIj[^IXNU,DIj[M0PWQNLNU,�0D�FWQNP+QNU,F+DtSBXN^IgIg(F+XNHzQNRIP+Q¯è@ù Ö�:%épÓ ù ¹�¼½: SBklRIH�LNH ¼ U*X+� � �

P+DIerg(F+XNU,QNU*`+HqeIH#�IDIU,QNH 'zÿ ^IgIg(F+XNHaSIU,DrP+eIeIU,QNU*F+DtSIQNR�PWQ � ÒyU,X�U,eIH�DaQNU,�IH#erklU*QNRrQNRIH|Ó+QNRmZ#F+V*^IM0D~F+J�QNR�H¯� � �M0P+QNLNU*� ° StXNF~QNR�PWQ É Ò � Ó Ù�°<¹z¼½°{Û Ò � '�� RIH#D s: Ò Ó Ù�°�¹�¼½:�Û Ò+S s: Ò Ó ÙNÙ�°<¹z¼½°�ÛKº Ï °<¹z¼ o Û Ò+S�P+DIes:Ó�°�Ù�° ¹ ¼½°�Û º Ï ° ¹ ¼ oSaP�J@P+M0U*V,U,PWLxH��!gILNH#XNXNU*F+DmJfLNFWM½V,U*D�H�P+LlM0F!eIH#V*X 'ó F+QNH}QNR�PWQ °½º�ÓºÙ�°�¹z¼½°�ÛKº Ï °<¹z¼ XNP+QNU,XN�IH�X�QNRIHmMU,DIU,M0PWVxZ�F+DIe�U*QNU,F+D�JfF+LP�j+H�DIH#LNP+V*U,��H#e�U,Da`+H�LNXNH

F+J ° SpXNFmk:H0M0P&b~klLNU,QNH s:�Ó °�° ºw: klU,QNR °/° º PWX�P+D�H#�!gIV*U,Z�U,Q�gILNF&¼NH�Z#QNU*F+D�M0PWQNLNU,� '[� RIU,X�gIP+LNQNU,Z�^IV,P+LZ=R�FWU,Z�HF+JvjWH#DIH�LNP+V,U*�#H�ehU,Da`+H�LNXNHb!U,H�V,eIXzQNRIHF+LNQNRIF+j+FWD�PWVES(F+LyXNH�V,J@�>Z#FWD|¼·^Ij+P+QNHaS(g�LNF&¼·H#Z�QNU,F+Dt�lFWQNR�H�LzZ�RIF+U*Z#H�XyF+Jj+H�DIH#LNP+V*U,��H#e}U*D!`WH#LNXNH�b!U,H�V,e}DIF+DI�>F+LNQNRIF+j+F+DIP+V_gILNF&¼NH�Z#QNU*F+DIXxF+D}QNF0QNRIH�XNP+MH�XNg�PWZ#H '� R�HlgILNH#Z�H#eIU*D�j�P+DIP+V,baXNU,XvP+XNXN^IM0H�XvQNR�PWQiQNRIHxZ�F+V,^IM0DIXvF+J ° P+LNHlV,U,DIH�P+LNV,b[U*DIe�H�g(H#DIeIH�D!QiXNF�QNR�PWQ °�¹�¼½°

U,XDIF+DI�>XNU,DIj+^IV*P+L ' »�F+L0P�`|PWLNU,H�QEbnF+JxLNH�P+XNF+DIX&SuQNRIU,XZ�F+DIeIU,QNU,FWD�U*X0JfLNH�\!^IH#DaQNV*bnDIF+Q[XNP+QNU,XN�IH�e�U,D�XNQNP+QNU,XNQNU*Z#P+Vk:F+LNd ' CED}F+LNeIH�LxQNF0MP+d+Hyg�LNFWj+LNH#XNX&S!U,QlU,XlDIP+QN^ILNP+V_QNF0Z#FWD�XNQNLN^IZ�QlQNRIH�M0PWQNLNU,�

´ Óc°{Ù�° ¹ ¼½°{Û º ° ¹ ¼JfFWLYXNFWM0HKZ=R�FWU,Z�HKFWJ�j+H�D�H�LNP+V*U,��H#e[U*D!`WH#LNXNHaS&klU,QNRGQNR�HTU*D!QNH�DaQNU,F+D[QNRIP+Q{´�o�XNR�FW^�V*ec(HTQNRIH:LNH#\!^IU*LNH#e[gILNF&¼NH�Z#QNU*F+DF+D[QNFyQNRIHKZ�F+V*^�MD0XNgIP+Z�HKF+J ° ' CEQYU,XuXNQNLNP+U*j+RaQNJfF+LNk:PWLNe0QNFzZ=RIH#Z=dqQNR�PWQY´�U*XuPzg�LNF&¼·H#Z�QNU,F+D[M0P+QNLNU*� Ù ´ Ò Ó ´ Û SU,JiP+DIehF+DIV,brU,JiQNR�H[j+H�D�H�LNP+V*U,��H#e�U,Da`+H�LNXNHU,XyP+DIDIU,RIU*V,P+QNU*D�j ' »�^�LNQNRIH�LNM0F+LNHaSwU*J °<¹z¼½° RIP+XyQNRIHXNPWM0HLNP+DIdP+X ° S�P+V*V�P+DIDIU,RIU*V,P+QNU*D�j�U*Da`+H#LNXNH�Xzb!U,H�V,e~QNR�HGXNP+M0H[gILNF&¼NH�Z�QNU,F+D~M0P+QNLNU*�_S�P+DIe�RIH�DIZ#H[QNRIHGXNP+M0H[�IQNQNH#e~`&P+V,^IHF+LlV*H#P+XNQN�>XN\a^�PWLNH#XxgILNF&¼NH�Z#QNU*F+D 'dSe�6f-47JL9|u� wi)dS.1Z5Mlf�6[Z5MlD!Nv\K01D38�Z5MlE�D3.1m_ö�H#Q¯oØc(HyPLNP+DIeIF+M½`WH#Z�QNF+LKU*D}P+DX���>e�U*M0H�D�XNU*F+DIP+V_P+�0DIHyXNgIP+Z�HzÍ '¤ H#�IDIH�g Ó r Ù o Û S ¡¡Ó Z#F&` Ù o Û SIP+DIe ¼ Ó¥¡Pº Ï S�klRIH�LNH ¡ U,X�QNP+d+H�D~QNF�c(HGg(F+XNU,QNU*`+HGeIH#�IDIU,QNH 'T� M0Fae�H�VXNg(H#Z�U,�IH�X�QNRIP+Q�gOV*U,H�X�U,D�P<�a�>eIU,M0H�DIXNU,F+DIP+V�M0P+DIU,J@F+V,eFg Ù � Û ��ÍqS:klU*QNRÇ�r`&P+LNb!U*DIjnU*D�P��!�>eIU,MH#DIXNU,FWD�PWVgIP+LNP+M0H�QNH�L]XNH#QO2 ' CED�jWH#DIH�LNP+VES ¡ e�H�g(H#DIeIXzFWD�gTSIP+DIe�RIH#DIZ�HGF+Dh� 'Kó F&kGS�o Ú g�U,X]P�`WH#Z�QNF+L�U,D�,}S�QNRIHQNP+DIj+H�DaQ�XNgIP+Z#HqF+JYÍqSIP+DIe ¼ U,X�P�M0H#QNLNU*ZqQNH#DIXNF+L�F+D�, '�ÿ U,DIZ�H ¼ eIH�gwH�DIeIX]F+D�gTS�k:HqeIH#DIF+QNHqc!b<è 8Ö� Lé @+SQNRIH�U*DID�H�LxgILNF!eI^IZ�Ql^�XNU*DIj�QNRIHyM0H#QNLNU*Z�QNH�DIXNF+L ¼ H�`&P+V,^IP+QNH�e}P+Qfg Ù � Û 'ötH�Q � Ò Ó�< gS� < � Ò JfF+LYÓ Ó ) Ö Þ�Þ�Þ Ö � ' CEDrM0P+QNLNU,��DIF+QNP+QNU*F+DtS%Æ Ó Ù`�|Ï´Ö Þ#Þ�Þ Ö���V!Û U*X�QNRIH�e�H�LNU,`&P+QNU*`+H�M0P+QNLNU*�F+JtQNR�HyM0H�P+DI�>`|PWV,^IH�`WH#Z�QNF+LlklU,QNR}LNH�XNg(H#Z�QlQNF0QNRIHygIP+LNP+M0H�QNH�LNXxP+Q�� 't� RIH#XNH�PWLNH�QNRIHyD�PWQN^�LNPWV_cIP+XNU,Xx`+H�Z�QNF+LNX

U,D5,1@aSBQNRIH�QNP+DIj+H#DaQ:XNg�PWZ#H]P+Q-�aSBklRIU,Z=R�M0P5b�cwH�LNH�j+P+LNeIH#e�P+X:P[XN^IcIXNgIP+Z#H]F+J , 't� RIH]`+H�Z#QNFWL-R Õ ,Qã@ klU,QNRZ�F+M0g(F+DIH#DaQNXR)Ò Ó è � Ò Ö o Ú g é @ Ó�Ù Æ ¹ ¼ Ù o Ú g ÛNÛ Ò

U,Xxd!DIF&klD}P+XxQNRIH�\!^IP+XNU*�>V,U,dWH#V*U,RIF!F!emH#XNQNU*M0P+QNU*D�j0Jf^IDIZ�QNU,F+DtS!F+Lx\a^�PWXNU,�>V,U*d+H�V,U,RIF!FaerXNZ�F+LNH�`+H�Z#QNFWL 'Y� RIH�LNF!F+QlF+JQNRIH[H�\!^IP+QNU*F+DhR)Ò Ùts� Û�Ó � U,X]QNR�Hq\!^IP+XNU,�>V*U,d+H�V,U*R�FaF!e�H�XNQNU,M0P+QNFWL&�TU,JuU,Q]H��!U*XNQNX&S�U,Q�M0P&bmDIF+Q]c(HG^ID�U*\!^IH 'xó FWQNHQNRIP+Q É Ò � R � P+LNH�QNRIHyZ�F+M0g(F+DIH�DaQNXlF+JtQNRIH�gILNF&¼NH�Z#QNU*F+D}FWJ+o Ú g¡F+D}QNF.,1@zklU,QNR}LNH�XNg(H#Z�QKQNF0QNRIHycIP+XNU,X ë� Ò ï '� RIU,XlU,XxQNRIHy�ILNXNQ ó H�klQNF+D}XNQNH�gmQNF&k:P+LNeIX s� '

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