aplicacions lineals

f(x,y) = 2x+4y

f(a+b,c+d) = 2·(a+b)+4·(c+d) = (2a+4c)+(2b+4d) = f(a,c)+f(b,d)

2x+4y = 0 <==> x = (-2)·y

f(x,y) = x^{2}·y^{4}

f(a·b,c·d) = (a·b)^{2}·(c·d)^{4} = (a^{2}·c^{4})·(b^{2}·d^{4}) = f(a,c)·f(b,d)

x^{2}·y^{4} = 1 <==> x = y^{(-2)}

f(x,y) = 3x+y

f(a+b,c+d) = 3·(a+b)+(c+d) = (3a+c)+(3b+d) = f(a,c)+f(b,d)

3x+y = 0 <==> x = (-1)·(1/3)·y

f(x,y) = x^{3}·y

f(a·b,c·d) = (a·b)^{3}·(c·d) = (a^{3}·c)·(b^{3}·d) = f(a,c)·f(b,d)

x^{3}·y = 1 <==> x = y^{(-1)·(1/3)}

f(x,y) = mx+ny

f(a+b,c+d) = m·(a+b)+n·(c+d) = (ma+nc)+(mb+nd) = f(a,c)+f(b,d)

mx+ny = 0 <==> x = (-1)·(n/m)·y

f(x,y) = x^{m}·y^{n}

f(a·b,c·d) = (a·b)^{m}·(c·d)^{n} = (a^{m}·c^{n})·(b^{m}·d^{n}) = f(a,c)·f(b,d)

x^{m}·y^{n} = 1 <==> x = y^{(-1)·(n/m)}

àlgebra lineal espais quocients

<a,b>+<c,d> = <a+c,b+d>

<a,b>·<c,d> = <a·c,b·d>

[ u ]_{F} = (u+(-v)) € i·<1,0>+j·<0,1> <==> ...

... (u+(-u)) € 0·<1,0>+0·<0,1> & ...

... (v+(-u)) € (-i)·<1,0>+(-j)·<0,1> & ...

... (u+(-v)) = (u+(-w))+(w+(-v))

( u )_{F} = (u/v) € i·<1,0>+j·<0,1> <==> ...

... (u/u) € 1·<1,0>+1·<0,1> & ...

... (v/u) € (1/i)·<1,0>+(1/j)·<0,1> & ...

... (u/v) = (u/w)·(w/v)

[ u_{1}+...(n)...+u_{n} ]_{F} = [ u_{1} ]_{F}+...(n)...+[ u_{n} ]_{F}

(u_{1}+...(n)...+u_{n})+(-1)·(v_{1}+...(n)...+v_{n}) = ...

... (u_{1}+(-1)·v_{1})+...(n)...+(u_{n}+(-1)·v_{n})

( u_{1}·...(n)...·u_{n} )_{F} = ( u_{1} )_{F}·...(n)...·( u_{n} )_{F}

(u_{1}·...(n)...·u_{n})/(v_{1}·...(n)...·v_{n}) = ...

... (u_{1}/v_{1})·...(n)...·(u_{n}/v_{n})

[ s·u ]_{F} = s·[ u ]_{F}

(s·u)+(-1)·(s·v) = s·(u+(-v))

( u^{s} )_{F} = ( ( u )_{F} )^{s} 

(u^{s})/(v^{s}) = (u/v)^{s}

( s·u )_{F} = ( u )_{F} 

(s·u)/(s·v) = (u/v)

( <n·p,n·q> )_{F} = ( ( <p^{(1/m)},q^{(1/m)}> )_{F} )^{m}

( <n·p,n·q> )_{F} = ( n·<p,q> )_{F} = ( <p,q> )_{F} = ...

... ( ( <p^{(1/m)},q^{(1/m)}> )^{m} )_{F} = ( ( <p^{(1/m)},q^{(1/m)}> )_{F} )^{m}

intersecció y suma de espais vectorials

k·<1,1> = i·<a,b> + j·<b,a>

i = k·( 1/(a+b) )

j = k·( 1/(b+a) )

k·<1,1> + i·<a,b> + j·<b,a> = ( i+k·( 1/(a+b) ) )·<a,b> + ( j+k·( 1/(b+a) ) )·<b,a>

k·<1,1> = i·<a,b> + j·<c,a>

i = k·( 1/(a+b) )

j = k·( 1/(c+a) )

k·<1,1> + i·<a,b> + j·<c,a> = ( i+k·( 1/(a+b) ) )·<a,b> + ( j+k·( 1/(c+a) ) )·<c,a>

matrius canóniques

matriu:
(-a)     (a/2)  (-a)
(a/2)   (a/2)  (a/2)
(a/2)   (-a)    (a/2)


polinomi característic:
P(x) = (-1)·(x+a)·(-1)·(x+(-1)(a/2))·(-1)·(x+(-1)(a/2))+...
... (-1)·(-1)·(-a)·(a/2)·(x+a)+(-1)·(-1)·(-a)·(a/2)·(x+(-1)(a/2))+(-1)·(-1)·(a/2)·(a/2)·(x+(-1)(a/2))+...
... (a^{3}/8)+(a^{3}/2)


valor propi:
e^{(1/3)·pi·i}·a·(1/4)^{(1/3)}


ker de base canónica
( (-a)+(-1)·e^{(1/3)·pi·i}·a·(1/4)^{(1/3)} )·x+(a/2)·y+(-a)·z = 0
(a/2)·x+( (a/2)+(-1)·e^{(1/3)·pi·i}·a·(1/4)^{(1/3)} )·y+(a/2)·z = 0
(a/2)·x+(-a)·y+( (a/2)+(-1)·e^{(1/3)·pi·i}·a·(1/4)^{(1/3)} )·z = 0

matrius canóniques

matriu:
a   c
b (-a)


polinomi característic:
P(x) = x^{2}+(-1)( a^{2}+bc )


valor propi:
( a^{2}+bc )^{(1/2)}


ker de base canónica:
( a+(-1)(a^{2}+bc)^{(1/2)} )·x+c·y = 0
b·x+( (-a)+(-1)(a^{2}+bc)^{(1/2)} )·y = 0


ker cíclic de base canónica:
( (-a)+(-1)(a^{2}+bc)^{(1/2)} )·x+b·y = 0
c·x+( a+(-1)(a^{2}+bc)^{(1/2)} )·y = 0


base canónica cíclica:
< cb·( a+(a^{2}+bc)^{(1/2)} ),cb^{2} >
< cb^{2},cb·( a+(a^{2}+bc)^{(1/2)} ) >

matrius canóniques

matriu:
a   b
b (-a)


polinomi característic:
P(x) = x^{2}+(-1)( a^{2}+b^{2} )


valor propi:
( a^{2}+b^{2} )^{(1/2)}


ker de base canónica:
( a+(-1)(a^{2}+b^{2})^{(1/2)} )·x+b·y = 0
b·x+( (-a)+(-1)(a^{2}+b^{2})^{(1/2)} )·y = 0


ker cíclic de base canónica:
( (-a)+(-1)(a^{2}+b^{2})^{(1/2)} )·x+b·y = 0
b·x+( a+(-1)(a^{2}+b^{2})^{(1/2)} )·y = 0


base canónica cíclica:
< a+(a^{2}+b^{2})^{(1/2)},b >
< b,a+(a^{2}+b^{2})^{(1/2)} >


matriu canónica:
(a^{2}+b^{2})^{(1/2)}            1
           0                       (a^{2}+b^{2})^{(1/2)}

índex de matemàtiques

index de etiquetes matemàtiques

dependencia lineal

si < u , v , w > és linealment independent ==> < u+v , v+w , w+u > és linealment independent.


demostració:
a(u+v) + b(v+w) + c(w+u) = (a+c)·u+(b+a)·v+(c+b)·w
a = (-c) & b = (-a) & c = (-b)
a = 0 & b = 0 & c = 0


 < u+(-v) , v+(-w) , w+(-u) > és linealment dependent.


demostració:
a(u+(-v)) + b(v+(-w)) + c(w+(-u)) = 0 <==> ( a = k & b = k & c = k )

aplicacions lineals

f(x,y) = <ax+by,cx+dy>


ker(f):
ax+by=0
cx+dy=0


x+(b/a)y=0
(c/d)x+y=0


x+(b/a)y=0
0·x+(1+(-1)(c/d)(b/a))y=0


Si ( a=b & c=d )  ==> x+y=0 ==> ( ( x=1 & y=(-1) ) or ( x=(-1) & y=1 ) )
Si ( ad=cb )  ==> x+(b/a)y=0 ==> ( ( x=1 & y=( (-a)/b ) ) or ( x=( (-b)/a) ) &  y=1 )
Si ( ad!=cb )  ==> ( x=0 & y=0 )
( ad=cb ) <==> ((-d)/c) = ((-b)/a)
Im(f):
ax+by=1
cx+dy=1
si ad!=cb ==> ( y=(a+(-c))/(ad+(-1)cb) & x=(d+(-b))/(ad+(-1)cb) )
ax+by=n
cx+dy=n
si ad!=cb ==> ( y=n·(a+(-c))/(ad+(-1)cb) & x=n·(d+(-b))/(ad+(-1)cb) )
ax+by=m
cx+dy=n
si ad!=cb ==> ( y=(an+(-c)m)/(ad+(-1)cb) & x=(dm+(-b)n)/(ad+(-1)cb) )


f(x,y) = <ax+by,cx+dy,ux+vy>


ker(f):
ax+by=0
cx+dy=0
ux+vy=0


x+(b/a)y=0
(c/d)x+y=0
(u/v)x+y=0


x+(b/a)y=0
0·x+(1+(-1)(c/d)(b/a))y=0
0·x+(1+(-1)(u/v)(b/a))y=0


Si ( a=b & c=d & u=v )  ==> x+y=0 ==> ( ( x=1 & y=(-1) ) or ( x=(-1) & y=1 ) )
Si ( ad=cb & av=ub)  ==> x+(b/a)y=0 ==> ( ( x=1 & y=( (-a)/b ) ) or ( x=( (-b)/a) ) &  y=1 )
Si ( ad!=cb or av!=ub )  ==> ( x=0 & y=0 )
( ad=cb & av=ub) <==> ( ((-d)/c) = ((-b)/a) & ((-v)/u) = ((-b)/a) )


Im(f):
ax+by=1+(-1)( (ud+(-1)cv)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
cx+dy=1+(-1)( (bu+(-1)av)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
ux+vy=1+(-1)( (ad+(-1)bc)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
si ( ad!=cb or ud!=cv or av!=bu ) ==>...
... ( y=(a+(-c)+u)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) &...
.... x=((-b)+d+(-v))/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )

aplicacions lineals

f(x,y) = ax+by


ker(f):
f(x,y)=0
ax+by=0
<x,y>=k·<(-b),a>=(-k)·<b,(-a)>=ak·<(-b)/a,1>
<x,y>=k·<(-b)/a,1>
<x,y>=k·<b,(-a)>=(-k)·<(-b),a>=bk·<1,(-a)/b>
<x,y>=k·<1,(-a)/b>


Im(f):
f(x,y)=v
ax+by=v
<x,y>=(v/2)·<(1/a),(1/b)>


f(x,y,z) = ax+by+cz


ker(f):
f(x,y,z)=0
ax+by+cz=0
<x,y,z>=k·<(-c),(-c),(a+b)>=(-k)·<c,c,(-1)(a+b)>=(-c)k·<(1,0,(-a)/c>+(-c)k·<(0,1,(-b)/c>
si ( x=1 & y=0 ) ==> <x,y,z>=k·<(1,0,(-a)/c>
si ( x=0 & y=1 ) ==> <x,y,z>=k·<(0,1,(-b)/c>
<x,y,z>=k·<(-b),(a+c),(-b)>=(-k)·<b,(-1)(a+c),b>=(-b)k·<(1,(-a)/b,0>+(-b)k·<(0,(-c)/b,1>
si ( x=1 & z=0 ) ==> <x,y,z>=k·<(1,(-a)/b,0>
si ( x=0 & z=1 ) ==> <x,y,z>=k·<(0,(-c)/b,1>
<x,y,z>=k·<(b+c),(-a),(-a)>=(-k)·<(-1)(b+c),a,a>=(-a)k·<((-c)/a),0,1>+(-a)k·<((-b)/a),1,0>
si ( y=0 & z=1 ) ==> <x,y,z>=k·<((-c)/a),0,1>
si ( y=1 & z=0 ) ==> <x,y,z>=k·<((-b)/a),1,0>


k·<(1,(-a)/b,0>=k·<(1,0,(-a)/c>+((-a)/b)k·<(0,1,(-b)/c>
k·<(0,(-c)/b,1>=0·k·<(1,0,(-a)/c>+((-c)/b)k·<(0,1,(-b)/c>
k·<((-c)/a),0,1>=((-c)/a)·k·<(1,0,(-a)/c>+0·k·<(0,1,(-b)/c>
k·<((-b)/a),1,0>=((-b)/a)·k·<(1,0,(-a)/c>+k·<(0,1,(-b)/c>


k·<(1,0,(-a)/c>=k·<(1,(-a)/b,0>+((-a)/c)k·<(0,(-c)/b,1>
k·<(0,1,(-b)/c>=0·k·<(1,(-a)/b,0>+((-b)/c)k·<(0,(-c)/b,1>
k·<((-c)/a),0,1>=((-c)/a)k·<(1,(-a)/b,0>+k·<(0,(-c)/b,1>
k·<((-b)/a),1,0>=((-b)/a)k·<(1,(-a)/b,0>+0·k·<(0,(-c)/b,1>


k·<(1,0,(-a)/c>=((-a)/c)k·<((-c)/a,0,1>+0·k·<((-b)/a,1,0>
k·<(0,1,(-b)/c>=((-b)/c)k·<((-c)/a,0,1>+k·<((-b)/a,1,0>
k·<(1,(-a)/b,0>=0·k·<((-c)/a,0,1>+((-a)/b)k·<((-b)/a,1,0>
k·<(0,(-c)/b),1>=k·<((-c)/a,0,1>+((-c)/b)k·<((-b)/a,1,0>






dim( ker(f) )=2


Im(f):
f(x,y,z)=v
ax+by+cz=v
<x,y,z>=(v/3)·<(1/a),(1/b),(1/c)>

polinomis y espai vectorial quocient

E = s·(x^{2}+1)+i·(x+1)+j·1
F = k·(x^{2}+x+1)


s·(x^{2}+1)+i·(x+1)+j·1+(-k)·(x^{2}+x+1) € [E]_{F}


(p+1)k·(x^{2}+1)+(p+1)k·(x+1)+((-p)+(-1))k·1+(-k)·(x^{2}+x+1) = pk·(x^{2}+x+1)
(p+1)k·(x^{2}+1)+(p+1)k·(x+1)+((-p)+(-1))k·1 = [ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1)
[ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1) = (p+1)k·(x^{2}+x+1)


(m+1)k·(x^{2}+1)+(n+1)k·(x+1)+(q+(-1))k·1+(-k)·(x^{2}+x+1) = mk·(x^{2}+1)+nk·(x+1)+qk·1
(m+1)k·(x^{2}+1)+(n+1)k·(x+1)+(q+(-1))k·1 = [ mk·(x^{2}+1)+nk·(x+1)+qk·1 ]+k·(x^{2}+x+1)


dim(E/F) = dim( [E]_{F} )+dim(F) =4
[0]_{F} = pk·(x^{2}+x+1)
[u]_{F} = mk·(x^{2}+1)+nk·(x+1)+qk·1
F=[0]_{F}+F
E/F=[u]_{F}+F


E = s·x^{2}+i·(x+1)+j·1
F = k·(x^{2}+x+1)


s·x^{2}+i·(x+1)+j·1 + (-k)·(x^{2}+x+1) € [E]_{F}




(p+1)k·x^{2}+(p+1)k·(x+1)+0·1 + (-k)·(x^{2}+x+1) = pk·(x^{2}+x+1)
(p+1)k·x^{2}+(p+1)k·(x+1)+0·1= [ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1)
[ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1) = (p+1)k·(x^{2}+x+1)


(m+1)k·x^{2}+(n+1)k·(x+1)+q·1 + (-k)·(x^{2}+x+1) = mk·x^{2}+nk·(x+1)+q·1
(m+1)k·x^{2}+(n+1)k·(x+1)+q·1 = [ mk·x^{2}+nk·(x+1)+q·1 ]+k·(x^{2}+x+1)




dim(E/F)=dim( [E]_{F} )+dim(F)=4
[0]_{F} = pk·(x^{2}+x+1)
[u]_{F} = mk·x^{2}+nk·(x+1)+q·1
F=[0]_{F}+F
E/F=[u]_{F}+F

espai vectorial quocient

E = i·<1,0>+j·<0,1>
F = k·<1,1>


i·<1,0>+j·<0,1>+(-k)·<1,1> € [E]_{F}


(p+1)k·<1,0>+(p+1)k·<0,1>+(-k)·<1,1> = pk·<1,1>
(p+1)k·<1,0>+(p+1)k·<0,1> = [ pk·<1,1> ]+k·<1,1>
[ pk·<1,1> ]+k·<1,1> = (p+1)k·<1,1>


(m+1)k·<1,0>+(n+1)k·<0,1>+(-k)·<1,1> = mk·<1,0>+nk·<0,1>
(m+1)k·<1,0>+(n+1)k·<0,1> = [ mk·<1,0>+nk·<0,1> ]+k·<1,1>


dim(E/F) = dim( [E]_{F} )+dim(F) =3
[0]_{F} = pk·<1,1>
[u]_{F} = mk·<1,0>+nk·<0,1>
F=[0]_{F}+F
E/F=[u]_{F}+F

polinomis y espai vectorial suma y intersecció

E = k·(x^{2}+x+1)
F = i·(x+1)+j·1


La suma
E+F = k·x^{2}+(k+i)·x+(k+i+j)·1
si k=0 ==>  E+F = i·(x+1)+j·1
si ( i=0 & j=0 ) ==>  E+F = k·(x^{2}+x+1)


La intersecció
k·(x^{2}+x+1) = i·(x+1)+j·1
0·(x^{2}+x+1) = 0·(x+1)+0·1
E[M]F = {0}




E = k·(x^{2}+x+1)
F = s·x^{2}+i·(x+1)+j·1


La suma
E+F = (k+s)·x^{2}+(k+i)·x+(k+i+j)·1
si k=0 ==>  E+F = s·x^{2}+i·(x+1)+j·1
si ( s=0 & i=0 & j=0 ) ==>  E+F = k·(x^{2}+x+1)


La intersecció
k·(x^{2}+x+1) = s·x^{2}+i·(x+1)+j·1
k·(x^{2}+x+1) = k·x^{2}+k·(x+1)+0·1
k·(x^{2}+x+1) = k·(x^{2}+x+1)
E[M]F = k·(x^{2}+x+1)


E = k·x
F = i·(x+1)+j·1


La suma
E+F = (k+i)·x+(i+j)·1
si k=0 ==>  E+F = i·(x+1)+j·1
si ( i=0 & j=0 ) ==>  E+F = k·x


La intersecció
k·x = i·(x+1)+j·1
k·x = k·(x+1)+(-k)·1
k·x = k·x
E[M]F = k·x




dim(E+F) = dim(E)+dim(F)+(-1)dim(E[M]F)

proyector 3d de gir horitzontal

k·<( cos(s)(c_{1}+(-1)a_{1})+sin(s)(c_{3}+(-1)a_{3}) ),(c_{2}+(-1)a_{2}),...
...( (-1)sin(s)(c_{1}+(-1)a_{1})+cos(s)(c_{3}+(-1)a_{3}) )>=...
...i·<cos(s),0,(-1)sin(s)>+j·<0,1,0>+<sin(s),0,cos(s)>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1}) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2}) )/(c_{3}+(-1)a_{3})


si s=0 ==>...
...k·<(c_{1}+(-1)a_{1}),(c_{2}+(-1)a_{2}),(c_{3}+(-1)a_{3})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<x,0,0>+j·<0,y,0>+<0,0,z> ]


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1}) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2}) )/(c_{3}+(-1)a_{3})


si s=(pi/2) ==>...
...k·<(c_{3}+(-1)a_{3}),(c_{2}+(-1)a_{2}),(-1)(c_{1}+(-1)a_{1})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<z,0,0>+j·<0,y,0>+<0,0,(-x)> ]


k=( (-1)/(c_{1}+(-1)a_{1}) )
i=( (-1)(c_{3}+(-1)a_{3}) )/(c_{1}+(-1)a_{1})
j=( (-1)(c_{2}+(-1)a_{2}) )/(c_{1}+(-1)a_{1})


si s=(-1)(pi/2) ==>...
...k·<(-1)(c_{3}+(-1)a_{3}),(c_{2}+(-1)a_{2}),(c_{1}+(-1)a_{1})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(-z),0,0>+j·<0,y,0>+<0,0,x> ]


k=( 1/(c_{1}+(-1)a_{1}) )
i=( (-1)(c_{3}+(-1)a_{3}) )/(c_{1}+(-1)a_{1})
j=( (c_{2}+(-1)a_{2}) )/(c_{1}+(-1)a_{1})










k·<f(s),g(s),d_{s}[f(s)]>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>


f(s) = cos(s)(c_{1}+(-1)a_{1})+sin(s)(c_{3}+(-1)a_{3})
d_{s}[f(s)] = (-1)sin(s)(c_{1}+(-1)a_{1})+cos(s)(c_{3}+(-1)a_{3})






si s=(pi/4) ==>...
...k·<(2^{(1/2)}/2)( (c_{1}+(-1)a_{1})+(c_{3}+(-1)a_{3}) ),...
...( (2^{(1/2)}/2)+(2^{(1/2)}/2) )(c_{2}+(-1)a_{2}),...
...(2^{(1/2)}/2)( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(2^{(1/2)}/2)x+(2^{(1/2)}/2)z,0,0>+...
...j·<0,(2^{(1/2)}/2)y+(2^{(1/2)}/2)y,0>+...
...<0,0,(2^{(1/2)}/2)z+(2^{(1/2)}/2)(-x)> ]


k=( 1/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) ) )
i=( ( (c_{1}+(-1)a_{1})+(c_{3}+(-1)a_{3}) ) )/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )
j=( 2(c_{2}+(-1)a_{2}) )/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )


si s=(pi/6) ==>...
...k·<( (1/2)(c_{1}+(-1)a_{1})+(3^{(1/2)}/2)·(c_{3}+(-1)a_{3}) ),...
...( (1/2)+(3^{(1/2)}/2) )(c_{2}+(-1)a_{2}),...
...( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(1/2)x+(3^{(1/2)}/2)z,0,0>+...
...j·<0,(1/2)y+(3^{(1/2)}/2)y,0>+...
...<0,0,(1/2)z+(3^{(1/2)}/2)(-x)> ]


k=( 1/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )
i=( ( (1/2)(c_{1}+(-1)a_{1})+(3^{(1/2)}/2)(c_{3}+(-1)a_{3}) ) )·...
(1/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) ) )
j=( ((1/2)+(3^{(1/2)}/2))(c_{2}+(-1)a_{2}) )/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )




En ser un gir hi ha una component ortogonal a les y.

proyector 3d frontal amb rotació

k·<(c_{1}+(-1)a_{1}),(c_{2}+(-1)a_{2}),(c_{3}+(-1)a_{3})>=...
...i·<cos(s),sin(s),0>+j·<(-1)sin(s),cos(s),0>+<0,0,1>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1})cos(s)+(c_{2}+(-1)a_{2})sin(s) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2})cos(s)+(c_{1}+(-1)a_{1})(-1)sin(s) )/(c_{3}+(-1)a_{3})

x·<1,0,0>=...
...( (c_{1}+(-1)a_{1})( cos(s) )^{2}+(c_{2}+(-1)a_{2})sin(s)cos(s) )/(c_{3}+(-1)a_{3})+...
...( (c_{2}+(-1)a_{2})(-1)sin(s)cos(s)+(c_{1}+(-1)a_{1})( sin(s) )^{2} )/(c_{3}+(-1)a_{3})


y·<0,1,0>=...
...( (c_{1}+(-1)a_{1})sin(s)cos(s)+(c_{2}+(-1)a_{2})( sin(s) )^{2} )/(c_{3}+(-1)a_{3})+...
...( (c_{2}+(-1)a_{2})( cos(s) )^{2}+(c_{1}+(-1)a_{1}(-1)sin(s)cos(s) )/(c_{3}+(-1)a_{3})

proyector frontal 3d

k·<c_{1}+(-1)a_{1},c_{2}+(-1)a_{2},c_{3}+(-1)a_{3}> =...
... i·<1,0,0>+j·<0,1,0>+<0,0,1>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1})/(c_{3}+(-1)a_{3}) )
j=( (c_{2}+(-1)a_{2})/(c_{3}+(-1)a_{3}) )


a_{i} = coordenades del observador.
c_{i} = coordenades del punt a observar.


k=( 1/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )
i=( ( (c_{1}+d_{1})+(-1)(a_{1}+d_{1}) )/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )
j=( ( (c_{1}+d_{2})+(-1)(a_{1}+d_{2}) )/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )

suma y intesecció de espais vectorials

E = k·<1,(-1)>
F = s·<(-1),1>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+k+(-s))·<1,0>+(j+s+(-k))·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<1,(-1)>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<(-1),1>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<1,(-1)> =  i·<1,0>+j·<0,1>
k·<1,(-1)> =  k·<1,0>+(-k)·<0,1>
k·<1,(-1)> =  k·<1,(-1)>
E[M]G = k·<1,(-1)>
La intersecció
s·<(-1),1> =  i·<1,0>+j·<0,1>
s·<(-1),1> =  (-s)·<1,0>+s·<0,1>
s·<(-1),1> =  s·<(-1),1>
F[M]G = s·<(-1),1>


La intersecció
k·<1,(-1)> =  s·<(-1),1>
k·<1,(-1)> =  (-k)·<(-1),1>
k·<1,(-1)> =  k·<1,(-1)>
E[M]F = k·<1,(-1)>


La intersecció
s·<(-1),1> =  k·<1,(-1)>
s·<(-1),1> =  (-s)·<1,(-1)>
s·<(-1),1> =  s·<(-1),1>
E[M]F = s·<(-1),1>


E[M]F[M]G = s·<(-1),1> = (-k)·<1,(-1)>
E[M]F[M]G = k·<1,(-1)> = (-s)·<(-1),1>


dim(E+F+G) = dim(E)+dim(F)+dim(G)+...
...+(-1)dim(E[M]F)+(-1)·dim(E[M]G)+(-1)·dim(F[M]G)+dim(E[M]F[M]G)


E = k·<2n,2n+1>
F = s·<2n+1,2n>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+2nk+(2n+1)s)·<1,0>+(j+(2n+1)k+2ns)·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<2n,2n+1>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<2n+1,2n>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<2n,2n+1> =  i·<1,0>+j·<0,1>
k·<2n,2n+1> =  2nk·<1,0>+(2n+1)k·<0,1>
k·<2n,2n+1> =  k·<2n,2n+1>
E[M]G = k·<2n,2n+1>
La intersecció
s·<2n+1,2n> =  i·<1,0>+j·<0,1>
s·<2n+1,2n> =  (2n+1)s·<1,0>+2ns·<0,1>
s·<2n+1,2n> =  s·<2n+1,2n>
F[M]G = s·<2n+1,2n>

suma y intersecció de espais vectorials

E = k·<1,1>
F = i·<1,0>+j·<0,1>


k·<1,1>+j·<0,1> = k·<1,0>+(j+k)·<0,1>
k·<1,1>+i·<1,0> = k·<0,1>+(i+k)·<1,0>


La suma
E+F = (i+k)·<1,0>+(j+k)·<0,1>
si k=0 ==> F+G = i·<1,0>+j·<0,1>
si ( i=0 & j=0 ) ==> F+G = k·<1,1>




La intersecció
k·<1,1> =  i·<1,0>+j·<0,1>
k·<1,1> =  k·<1,0>+k·<0,1>
k·<1,1> =  k·<1,1>
E[M]F = k·<1,1>


dim(E+F) =dim(E)+dim(F)+(-1)·dim(E[M]F)


E = k·<1,0>
F = s·<0,1>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+k)·<1,0>+(j+s)·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<1,0>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<0,1>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<1,0> =  i·<1,0>+j·<0,1>
k·<1,0> =  k·<1,0>+0·<0,1>
k·<1,0> =  k·<1,0>
E[M]G = k·<1,0>
La intersecció
s·<0,1> =  i·<1,0>+j·<0,1>
s·<0,1> =  0·<1,0>+s·<0,1>
s·<0,1> =  s·<0,1>
F[M]G = s·<0,1>


dim(E+F+G) = dim(E)+dim(F)+dim(G)+...
...+(-1)dim(E[M]F)+(-1)·dim(E[M]G)+(-1)·dim(F[M]G)+dim(E[M]F[M]G)

producte escalar

<k,0>[o]<0,k> = 0
<(-k),0>[o]<0,(-k)> = 0


<k,0>[o]<0,(-k)> = 0
<(-k),0>[o]<0,k> = 0


<k,k>[o]<k,(-k)> = ( k^{2}+(-1)k^{2}) = 0
<k,k>[o]<(-k),k> = ( (-1)k^{2}+k^{2}) = 0


<(-k),(-k)>[o]<k,(-k)> = ( (-1)k^{2}+k^{2}) = 0
<(-k),(-k)>[o]<(-k),k> = ( k^{2}+(-1)k^{2}) = 0


<k,k>[o]<k,k> = ( k^{2}+k^{2} ) = 2k^{2}
<k,k>[o]<(-k),(-k)> = (-1)( k^{2}+k^{2} ) = (-2)k^{2}


<k,k>[o]<0,k> = k^{2}
<k,k>[o]<k,0> = k^{2}


<(-k),(-k)>[o]<0,k> = (-1)k^{2}
<(-k),(-k)>[o]<k,0> = (-1)k^{2}


<k,k>[o]<0,(-k)> = (-1)k^{2}
<k,k>[o]<(-k),0> = (-1)k^{2}


<(-k),(-k)>[o]<0,(-k)> = k^{2}
<(-k),(-k)>[o]<(-k),0> = k^{2}