Interpreta\303\247\303\243o geom\303\251trica de transforma\303\247\303\265es lineares
Um dos t\303\263picos mais importantes em \303\201lgebra Linear \303\251 a \303\241lgebra matricial. Ela aparece como representa\303\247\303\265es de operadores lineares e de transforma\303\247\303\265es lineares entre espa\303\247os vetoriais de dimens\303\243o finita, frequente nas aplica\303\247\303\265es. A versatilidade do Maple V \303\251 muito \303\272til n\303\243o somente para efetuar as opera\303\247\303\265es matriciais, mas tamb\303\251m para permitir uma interpreta\303\247\303\243o geom\303\251trica dos efeitos dos movimentos mais importantes do plano e do espa\303\247o, atrav\303\251s de seus recursos gr\303\241ficos. Em geral, num primeiro curso de \303\201lgebra Linear as propriedades geom\303\251tricas de transforma\303\247\303\265es lineares n\303\243o s\303\243o suficientemente compreendidas, e a utiliza\303\247\303\243o de Maple V pode auxiliar neste sentido. Os recursos de anima\303\247\303\243o s\303\243o \303\272teis para explorar os movimentos e perceber o car\303\241ter isom\303\251trico de alguns dos mais importantes movimentos do plano e do espa\303\247o.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 de movimentos atrav\303\251s da \303\241lgebra matricial
Um estudo completo de movimentos do plano e do espa\303\247o, assim como da \303\241lgebra matricial de transforma\303\247\303\265es lineares, utilizando os comandos do pacote linalg de \303\201lgebra Linear, pode ser encontrada em [B-P]. Nesta apresenta\303\247\303\243o, vamos ilustrar apenas alguns exemplos.
Reflex\303\243o no plano, segundo o eixo y
\303\211 a transforma\303\247\303\243o dada por :
Ry : R\302\262 - R\302\262
Ry ( x , y ) = ( -x , y )
Exemplo: Ry ( 1 , 1 ) = ( -1 , 1 ).
Este operador tem a sua representa\303\247\303\243o matricial dada por:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEjUnlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGKy1GLDYtUSomY29sb25lcTtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGU0YrLUZHNiVRJ21hdHJpeEYnRkpGTS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZZNiYtRiM2Jy1GWTYmLUYjNigtRiw2LVEqJnVtaW51czA7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRmNvLUkjbW5HRiQ2JFEiMUYnRi8tRiw2LVEiLEYnRi9GMi9GNkZMRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZmbzYkUSIwRidGLy8lK2V4ZWN1dGFibGVHRjRGL0YvLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmlvLUZZNiYtRiM2J0ZfcEZpb0Zlb0ZicEYvRi9GZHBGZ3BGYnBGL0YvRmRwRmdwRmJwRi9GLy1GLDYtUSI7RidGL0YyRlxwRjdGOUY7Rj1GP0ZBRlQtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0ZmcS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GYnE2JkZkcUZncUZpcS9GXHJRJWF1dG9GJ0ZhcUZeci1GRzYjUSFGJ0ZicEYvComo exemplo, seja v o vetor sobre o qual ser\303\241 aplicada a transforma\303\247\303\243o linear Ry.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Se q \303\251 o vetor Ry ( v ), o comando evalm(Ry &* v) efetua a multiplica\303\247\303\243o matricial de Ry por vetor coluna v: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Vamos considerar P e Q , pontos do plano, tal que OP e OQ representem os vetores v e q :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O comando abaixo mostra o vetor antes da aplica\303\247\303\243o de Ry e ap\303\263s a aplica\303\247\303\243o de Ry.
. v antes de T......................: cor vermelha
. q = Ry( v ) depois de Ry ...: cor azul
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Proje\303\247\303\243o Segundo uma Reta qualquer passando pela Origem
Exemplo:
Consideremos a reta y = 3x . Um vetor dire\303\247\303\243o desta reta pode ser dado por ( 1, 3 ), que \303\251 invariante por proje\303\247\303\243o sobre a reta dada. Claramente, o vetor ( -3 , 1 ) \303\251 ortogonal ao vetor ( 1, 3 ), constitui uma base de R\302\262 juntamente com ( 1 , 3 ) e a proje\303\247\303\243o sobre a reta dada leva ao vetor nulo .
Temos duas bases a considerar:
Base can\303\264nica : { ( 1 , 0 ) , ( 0 , 1 ) }
Base B : { ( 1 , 3 ) , ( -3 , 1 ) }
B C B'
A proje\303\247\303\243o ter\303\241 como matriz de transforma\303\247\303\243o : I ] * T ] * I ] , onde est\303\243o envolvidas as matrizes mudan\303\247a de base.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 matriz T representa a proje\303\247\303\243o, relativa \303\240 base B. Observamos que ela mant\303\251m inva riante o vetor ( 1 , 3 ) e anula o vetor ( -3 , 1 ) ortogonal a ele.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mudanca_base2 : matriz de mudan\303\247a da base B para a Can\303\264nica .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mudanca_base1 : matriz de mudan\303\247a da base Can\303\264nica para a B .
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Ent\303\243o, a matriz procurada \303\251 dada por :
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEubWF0cml6X3RyYW5zZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRJmV2YWxtRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNi0tRiw2JVEubXVkYW5jYV9iYXNlMkYnRi9GMkY1LUY2Ni1RIyYqRidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUYsNiVRIlRGJ0YvRjJGNUZqbkY1LUYsNiVRLm11ZGFuY2FfYmFzZTFGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRkpGVEZjb0Y5Por exemplo, consideremos o vetor ( 1 , 1 ) . Aplicando a matriz acima obtida, teremos o vetor v , que \303\251 a proje\303\247\303\243o de ( 1 , 1 ) sobre a reta dada .LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjdjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSd2ZWN0b3JGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JS1GWTYmLUYjNictSSNtbkdGJDYkUSIxRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRidGW28vJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zlb0Y5RjktRjY2LVEiO0YnRjlGO0Zib0ZARkJGREZGRkhGSkZURmVvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjdjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSd2ZWN0b3JGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JS1GWTYmLUYjNictSSNtbkdGJDYkUSIxRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRidGW28vJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zlb0Y5RjktRjY2LVEiO0YnRjlGO0Zib0ZARkJGREZGRkhGSkZURmVvRjk=Podemos visualizar graficamente o exemplo, atrav\303\251s de :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Vetor ( 1 , 1 ) ......... representado pela cor verde .
Vetor( 2/5 , 6/5 ) ........ representado pela cor azul .
Reta y = 3x ........ representada pela cor vermelha .
Podemos obter a imagem de qualquer vetor do plano pela transforma\303\247\303\243o, repetindo o procedimento.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 de utiliza\303\247\303\243o de recursos de anima\303\247\303\243o no Estudo de Movimentos: Cisalhamento
Iremos apresentar uma pequena amostra dos recursos gr\303\241ficos e de anima\303\247\303\243o que o Maple V possui, aplicados no estudo de operadores lineares do plano. A t\303\251cnica da \303\241lgebra matricial para representar os movimentos continua sendo a base te\303\263rica fundamental. A programa\303\247\303\243o da anima\303\247\303\243o foi realizada pela bolsista do REENGE Karina Moreno Passos. Os recursos gr\303\241ficos s\303\243o excelentes para compreender as propriedades geom\303\251tricas das matrizes de movimentos.
Descri\303\247\303\243o
Esta anima\303\247\303\243o gr\303\241fica consiste em cisalhar um quadrado , sendo que :
- O quadrado \303\251 definido no procedimento , ou seja , a figura \303\251 pr\303\251-fixada .
- A anima\303\247\303\243o \303\251 obtida atrav\303\251s de quadros ( frames ) e o fator de cisalhamento (FC) adotado \303\251 2 .Detalhes do Procedimento
. Nome do Procedimento : CisalhaQuadr2D( )
. Chamada do procedimento : CisalhaQuadr2D( );
. Par\303\242metros para a chamada do procedimento : nenhum
. Bibliotecas do Maple utilizadas : < linalg e < plots
. Linguagem de programa\303\247\303\243o utilizada : linguagem pr\303\263pria do Maple
. N\303\272mero de frames : 5
Defini\303\247\303\243o do c\303\263digo de 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Exemplos de anima\303\247\303\243o de figuras com Movimentos de rota\303\247\303\243o
Vamos apresentar aqui um exemplo de utiliza\303\247\303\243o do movimento de rota\303\247\303\243o, para animar pontos, pol\303\255gonos e curvas parametrizadas.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A matriz de rota\303\247\303\243o de \303\242ngulo \316\270 radianos medido positivamente no sentido anti-hor\303\241rio \303\251 dada 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que \303\251 uma matriz 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Veja os resultados: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Agora, uma fun\303\247\303\243o que a cada ponto P do plano associa o ponto Q = RotP(P,t) que \303\251 o ponto P rotacionado de \303\242ngulo t em torno da origem: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 convers\303\243o para 'list' foi feita para apresentar os pontos em formato de 'list' e n\303\243o de 'array' como seria o resultado de evalm . Matrizes e vetores s\303\243o 'array's.
Vejamos agora o resultado da fun\303\247\303\243o: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Vamos agora aos desenhos:
Iniciemos com uma rotina para desenhar um ponto escolhido P girar em torno da origem: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 definimos os v\303\251rtices P1, P2, P3, P4 de um pol\303\255gono: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, vamos girar esse pol\303\255gono de \303\242ngulo Abertura em torno da origem: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Agora, vamos girar uma curva parametrizada em torno da 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T\303\263picos e Coment\303\241rios Finais
O pacote de \303\201lgebra Linear do Maple V \303\251 bastante completo e \303\272til. Nesta apresenta\303\247\303\243o, n\303\243o podemos apresentar exemplos de todos os seus recursos, mas al\303\251m dos exemplos mostrados acima, em [B-P] temos preparado material sobre a linguagem b\303\241sica de \303\201lgebra Linear, teoria geral de transforma\303\247\303\265es lineares, auto-valores e problemas de diagonaliza\303\247\303\243o, produto interno e operadores geom\303\251tricos. Um anexo com as aplica\303\247\303\265es tamb\303\251m est\303\241 em prepara\303\247\303\243o. Um ponto a ser observado tamb\303\251m na utiliza\303\247\303\243o de Softwares \303\251 a sua limita\303\247\303\243o, envolvendo problemas de aproxima\303\247\303\243o nos resultados, ou ainda a interpreta\303\247\303\243o das respostas que o Software nos apresenta de acordo com o problema que n\303\263s queremos resolver. Acreditamos que temos um vasto campo de estudo para ser explorado.