álgebra lineal

(f+g)(x) = f(x)+g(x)

(f·g)(x) = f(x)·g(x)

(s·f)(x) = s·f(x)

(f^{s})(x) = ( f(x) )^{s}

( s·(f+g) )(x) = s·( (f+g)(x) ) = s·( f(x)+g(x) ) = ...

... s·f(x)+s·g(x) = (s·f)(x)+(s·g)(x) = ( (s·f)+(s·g) )(x)

s·(f+g) = (s·f)+(s·g)

( (f·g)^{s} )(x) = ( (f·g)(x) )^{s} = ( f(x)·g(x) )^{s} = ...

... ( f(x) )^{s}·( g(x) )^{s} = f^{s}(x)·g^{s}(x) = ( f^{s}·g^{s} )(x)

(f·g)^{s} = f^{s}·g^{s}

simétrica de suma:

v(-x) = v(x)

anti-simétrica de suma:

u(-x) = (-1)·u(x)

simétrica de producto:

v(1/x) = v(x)

anti-simétrica de producto:

u(1/x) = (1/u(x))

[Eu(x)][Ev(x)][ u(x) anti-simétrica de suma & v(x) simétrica de suma & ...

... f(x) = (1/2)·( u(x)+v(x) ) ].

Demostración:

f(x) = (1/2)·( f(x)+f(x) ) = ...

... (1/2)·( f(x)+(-1)·f(-x)+f(-x)+f(x) ) = ...

... (1/2)·( u(x) = ( f(x)+(-1)·f(-x) )+v(x) = ( f(-x)+f(x) ) )

dim(f(x)) = (1/2)·( dim(u(x))+dim(v(x)) )

[Eu(x)][Ev(x)][ u(x) anti-simétrica de producto & v(x) simétrica de producto & ...

... f(x) = ( u(x)·v(x) )^{(1/2)} ].

Demostración:

f(x) = ( f(x)·f(x) )^{(1/2)} = ...

... ( ( f(x)/f(1/x) )·( f(1/x)·f(x) ) )^{(1/2)} = ...

... ( u(x) = ( f(x)/f(1/x) )+v(x) = ( f(1/x)·f(x) ) )^{(1/2)}

dim(f(x)) = ( dim(u(x))·dim(v(x)) )^{(1/2)}

s·<a_{ij}+b_{ij}> = s·( <a_{ij}>+<b_{ij}> ) = s·<a_{ij}>+s·<b_{ij}>

<a_{ij}·b_{ij}>^{s} = ( <a_{ij}>·<b_{ij}> )^{s} = <a_{ij}>^{s}·<b_{ij}>^{s}

simétrica:

v_{ji} = v_{ij}

anti-simétrica de suma:

u_{ji} = (-1)·u_{ij}

anti-simétrica de producto:

u_{ji} = (1/u_{ij})

[E<u_{ij}>][E<v_{ij}>][ <u_{ij}> anti-simétrica de suma & <v_{ij}> simétrica & ...

... <a_{ij}> = (1/2)·( <u_{ij}>+<v_{ij}> ) ].

Demostración:

<a_{ij}> = (1/2)·( <a_{ij}>+<a_{ij}> ) = ...

... (1/2)·( <a_{ij}>+(-1)·<a_{ji}>+<a_{ji}>+<a_{ij}> ) = ...

... (1/2)·( <( a_{ij}+(-1)·a_{ji} )>+<( a_{ji}+a_{ij} )> ) = ...

... (1/2)·( <u_{ij} = ( a_{ij}+(-1)·a_{ji} )>+<v_{ij} = ( a_{ji}+a_{ij} )> )

dim(a_{ij}) = (1/2)·( dim(u_{ij})+dim(v_{ij}) ) = (1/2)·( n^{2}+n^{2} ) = n^{2}

[E<u_{ij}>][E<v_{ij}>][ <u_{ij}> anti-simétrica de producto & <v_{ij}> simétrica & ...

... <a_{ij}> = ( <u_{ij}>·<v_{ij}> )^{(1/2)} ].

Demostración:

<a_{ij}> = ( <a_{ij}>·<a_{ij}> )^{(1/2)} = ...

... ( <a_{ij}>·<(1/a_{ji})>·<a_{ji}>·<a_{ij}> )^{(1/2)} = ...

... ( <( a_{ij}/a_{ji} )>·<( a_{ji}·a_{ij} )> )^{(1/2)} = ...

... ( <u_{ij} = ( a_{ij}/a_{ji} )>·<v_{ij} = ( a_{ji}·a_{ij} )> )^{(1/2)}

dim(a_{ij}) = ( dim(u_{ij})·dim(v_{ij}) )^{(1/2)} = ( n^{2}·n^{2} )^{(1/2)} = n^{2}

d_{x}[ ae^{2x}+be^{x} ] = 2ae^{2x}+be^{x}

( <2,0>,<0,1> )[o]<a,b> = <2a,b>

int[ ae^{2x}+be^{x} ] d[x] = (1/2)·ae^{2x}+(1/1)·be^{x}

( <(1/2),0>,<0,(1/1)> )[o]<a,b> = <(1/2)·a,(1/1)·b>

( <2,0>,<0,1> )[o]( <(1/2),0>,<0,(1/1)> ) = ( <1,(3/1)·0>,<(3/2)·0,1> )

( <(1/2),0>,<0,(1/1)> )[o]( <2,0>,<0,1> ) = ( <1,(3/2)·0>,<(3/1)·0,1> )

A[o]A^{[o](-1)} = [ a_{ii}=1 & a_{ij} = k_{ij}·0 ]

A^{[o](-1)}[o]A = [ a_{ii}=1 & a_{ji} = k_{ij}·0 ]

( <a,b>,<b,a> )[o]( ( 1/det(A) )·( <a,(-b)>,<(-b),a> ) ) = ...

... ( 1/det(A) )·( <a^{2}+(-1)·b^{2},(-1)·ab·0>,<ab·0,a^{2}+(-1)·b^{2}> )

( ( 1/det(A) )·( <a,(-b)>,<(-b),a> ) )[o]( <a,b>,<b,a> ) = ...

... ( 1/det(A) )·( <a^{2}+(-1)·b^{2},ab·0>,<(-1)·ab·0,a^{2}+(-1)·b^{2}> )