álgebra lineal: sub-espais y combinacions lineals

Axioma:

[As][ s€K ==> s·0 = 0 ]

<0,x>+<0,y> = <2·0,x+y> = <0,x+y>

s·<0,x> = <s·0,s·x> = <0,s·x>

i·<0,x>+j·<0,x> = <0,0> ==> ( i = i & j = (-i) ) || ( i = (-j) & j = j )

i·<0,x>+j·<0,x> = <0,x> ==> ( i = a & j = 1+(-a) ) || ( i = 1+(-a) & j = a )

i·<0,x>+j·<0,y> = <0,0> ==> ( i = (y/x) & j = (-1) ) || ( i = (-1) & j = (x/y) )

i·<0,x>+j·<0,y> = <0,k> ==> ...

... ( i = (1+(-a))·(k/x) & j = a·(k/y) ) || ( i = a·(k/x) & j = (1+(-a))·(k/y) )

Axioma:

[As][ s€K ==> 1^{s} = 1 ]

<1,x>·<1,y> = <1^{2},x·y> = <1,x·y>

<1,x>^{s} = <1^{s},x^{s}> = <1,x^{s}>

<1,x>^{i}·<1,x>^{j} = <1,1> ==> ( i = i & j = (-i) ) || ( i = (-j) & j = j )

<1,x>^{i}·<1,x>^{j} = <1,x> ==> ( i = a & j = 1+(-a) ) || ( i = 1+(-a) & j = a )

<1,x>^{i}·<1,y>^{j} = <1,1> ==> ( i = log_{x}(y) & j = (-1) ) || ( i = (-1) & j = log_{y}(x) )

<1,x>^{i}·<1,y>^{j} = <1,k> ==> ...

... ( i = (1+(-a))·log_{x}(k) & j = a·log_{y}(k) ) || ( i = a·log_{x}(k) & j = (1+(-a))·log_{y}(k) )