Real Analisys - MINIPROYECTO DE ALGEBRA LINEAL 2 Y...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MINIPROYECTO DE ALGEBRA LINEAL 2 Y CUADRATICA 26 de marzo de 2011 1. El espacio vectorial Z n 2 Se consedera el conjunto Z n 2 = { x = ( x 1 , x 2 , . . . , x n ) /x i Z 2 } y el campo escalar Z 2 = { 1 , 2 } . a. Demostrar que Z n 2 es un espacio vectorial. La suma esta definida as´ ı : 0+0=0 0+1=1 1+0=1 1+1=0 El producto esta definido as´ ı : 0x0=0 0x1=0 1x0=0 1x1=1 Sean x,y,z Z n 2 y α, β Z 2 ; con x, y, z de la forma: x = ( x 1 , x 2 , . . . , x n ) , y = ( y 1 , y 2 , . . . , y n ) y z = ( z 1 , z 2 , . . . , z n ) para x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n , z 1 , z 2 , . . . , z n Z 2 1. x, y Z n 2 , ( x + y ) Z n 2 x + y = ( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) = ( x 1 + y 1 | {z } Z 2 , x 2 + y 2 | {z } Z 2 , . . . . . . , x n + y n | {z } Z 2 | {z } Z n 2 ) Suma en Z n 2 Entonces : ( x + y ) Z n 2 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2. x, y Z n 2 ,(x+y)+z=x+(y+z) ( x + y ) + z = [( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n )] + ( z 1 , z 2 , . . . , z n ) = ( x 1 + y 1 , x 2 + y 2 , . . . . . . , x n + y n ) + ( z 1 , z 2 , . . . , z n ) Suma en Z n 2 = ( x 1 + y 1 + z 1 , x 2 + y 2 + z 2 , . . . . . . , x n + y n + z n ) Suma en Z n 2 = [ x 1 + ( y 1 + z 1 ) , x 2 + ( y 2 + z 2 ) , . . . . . . , x n + ( y n + z n )] Asociatividad en Z 2 = ( x 1 , x 2 , . . . , x n ) + ( y 1 + z 1 , y 2 + z 2 , . . . . . . , y n + z n ) Suma en Z n 2 = ( x 1 , x 2 , . . . , x n ) + [( y 1 , y 2 , . . . , y n ) + ( z 1 , z 2 , . . . , z n )] Suma en Z n 2 = x + ( y + z ) 3. 0 Z n 2 tl que x Z n 2 se tiene que x + 0 = 0 + x = x . Sea 0 = (0 , 0 , . . . , 0) con 0 Z 2 0 + x = (0 , 0 , . . . , 0) + ( x 1 , x 2 , . . . , x n ) = (0 + x 1 , 0 + x 2 , . . . . . . , 0 + x n ) Suma en Z n 2 = ( x 1 , x 2 , . . . , x n ) Suma en Z n 2 = x 4. x Z n 2 , x * Z n 2 tal que x + x * = 0 Sea x * = x x + x * = ( x 1 , x 2 , . . . , x n ) + ( x 1 , x 2 , . . . , x n ) = ( x 1 + x 1 , x 2 + x 2 , . . . . . . , x n + x n ) Suma en Z n 2 = (0 , 0 , . . . , 0) Suma en Z 2 = 0 5. x, y Z n 2 tal que x + y = y + x x + y = ( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) = ( x 1 + y 1 , x 2 + y 2 , . . . . . . , x n + y n ) Suma en Z n 2 = ( y 1 + x 1 , y 2 + x 2 , . . . . . . , y n + x n ) Conmutatividad en Z 2 = y + x Suma en Z n 2 6. x Z n 2 α Z 2 , tal que ( αx ) Z n 2 αx = α ( x 1 , x 2 , . . . , x n ) = ( αx 1 |{z} Z 2 , αx 2 |{z} Z 2 , . . . . . . , αx n |{z} Z 2 | {z } Z n 2 ) Produncto en Z n 2 Entonces: ( αx ) Z n 2 7. x, y Z n 2 , α Z 2 , tal que α ( x + y ) = αx + αy α ( x + y ) = α [( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n )] = α ( x 1 + y 1 , x 2 + y 2 , . . . . . . , x n + y n ) Suma en Z n 2 = [ α ( x 1 + y 1 ) , α ( x 2 + y 2 ) , . . . . . . , α ( x n + y n ) Producto en Z n 2 ] = ( αx 1 + αy 1 , αx 2 + αy 2 , . . . . . . , αx n + αy n ) Distributiva en Z 2 = ( αx 1 αx 2 , . . . , αx n ) + ( αy 1 , αy 2 , . . . , αy n ) Suma
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern