Efrita

El espacio n-dimensional dado por x=x1,x2,x3…….xn=fn=V con F un campo escalar, verificar si es un espacio vectorial.
i) x+y=y+x ,x,y=εFn=V
SEAN x=x1,x2,x3…….xn
y=y1,y2,y3…….yn
z=z1,z2,z3…….zn
α, β ε T
y+x=y1,y2,y3…….yn+x1,x2,x3…….xn=x1+y1,x2+y2,x3+y3…….xn+yn
=x1,x2,x3…….xn+y1,y2,y3…….yn
=x+y

ii)x+y+z=x+(y+z)
x+y+z=x1,x2,x3…….xn+y1,y2,y3…….yn+z1,z2,z3…….zn
=x1+y1,x2+y2,x3+y3…….xn+yn+(z1,z2,z3…….zn)=x1+y1+z1),x2+(y2+z2),x3+(y3+z3)…….xn+(yn+zn
=(x1,x2,x3…….xn)+y1+z1,y2+z2,y3+z3…….yn+zn
=x1,x2,x3…….xn+y1,y2,y3…….yn+(z1,z2,z3…….zn)
=x+(y+z) ∴x+y+z=x+(y+z)

iii) x+0=x1,x2,x3…….xn+01,02,03…….0n
=x1+01,x2+02,x3+03…….xn+0n
=x1,x2,x3…….xn=x ∴ x+0= x

iv) SEA0=01,02,03…….0n
Y 0=xi-xi con i= 1,2……n
01,02,03…….0n=(x1-x1, x2-x2,……xn-xn)
=x1+-x1, x2+-x2,……xn+(-xn)=x1+y1,x2+y2,…….xn+yn
=x+y
0=x+y
y=-x1,-x2,…….-xn

v)αβx=α(βx)
=αβ(x1,x2,xn)

=α*β*x1α,βx2(α,βxn)
=(αβ1,x1 α2β2,x2αn(βn,xn)
=α1,α2,αnβ1,β2,βn(x1,x2,xn)
=αβ(x)

vi) αx+y=αx+αy
=αx1,x2,,xn
=αx1,x2,xn, +αy1,αy2,αyn,
=αx+αy

vii) α+βx=αx+βx
=[αβx1,x2,xn
=[(α)(x1,x2,xn)+(β)(x1,x2,xn)
=α,x1α,x2α,xn+β,x1β,x2(β,xn)
= αx+βx
viii) 1x=x