En matemàtiques , una aplicació lineal és un morfisme entre dos espais vectorials que respecta l'operació suma de vectors i la multiplicació escalar definides en aquests espais vectorials, o, en altres paraules que preserven les combinacions lineals .
Sigui
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
aplicació on
E
{\displaystyle \mathbf {E} }
F
{\displaystyle \mathbf {F} }
K
{\displaystyle \mathbb {K} }
Una aplicació que compleixi la primera condició es diu additiva , si, en canvi compleix la segona es diu homogènia .
Si
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
aplicació lineal ,
∀
x
,
y
∈
E
{\displaystyle \forall x,y\in \mathbf {E} }
∀
a
,
b
∈
K
{\displaystyle \forall a,b\in \mathbb {K} }
f
(
a
x
+
b
y
)
=
a
f
(
x
)
+
b
f
(
y
)
{\displaystyle f(ax+by)=af(x)+bf(y)\,}
f
(
∑
i
=
1
m
a
i
x
i
)
=
∑
i
=
1
m
a
i
f
(
x
i
)
{\displaystyle f\left(\sum _{i=1}^{m}a_{i}x_{i}\right)=\sum _{i=1}^{m}a_{i}f(x_{i})}
f
(
0
→
)
=
0
→
{\displaystyle f({\vec {0}})={\vec {0}}}
f
(
−
x
)
=
−
f
(
x
)
{\displaystyle f(-x)=-f(x)\,}
Si
g
:
F
→
G
{\displaystyle g:\mathbf {F} \rightarrow \mathbf {G} }
aplicació lineal , aleshores:
g
∘
f
:
E
→
G
{\displaystyle g\circ f:\mathbf {E} \rightarrow \mathbf {G} }
aplicació lineal .
modifica
Siguin
E
{\displaystyle \mathbf {E} }
F
{\displaystyle \mathbf {F} }
{
u
1
,
…
,
u
n
}
{\displaystyle \{u_{1},\dots ,u_{n}\}}
{
v
1
,
…
,
v
m
}
{\displaystyle \{v_{1},\dots ,v_{m}\}}
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
f
{\displaystyle \ f}
f
(
u
1
)
,
…
,
f
(
u
n
)
{\displaystyle f(u_{1}),\dots ,f(u_{n})}
F
{\displaystyle \mathbf {F} }
f
(
u
i
)
=
∑
j
=
1
m
λ
i
j
v
j
,
i
=
1
,
…
,
n
{\displaystyle f(u_{i})=\sum _{j=1}^{m}\lambda _{i}^{j}v_{j},i=1,\dots ,n}
A
{\displaystyle \ A}
f
{\displaystyle \ f}
{
u
1
,
…
,
u
n
}
{\displaystyle \{u_{1},\dots ,u_{n}\}}
{
v
1
,
…
,
v
m
}
{\displaystyle \{v_{1},\dots ,v_{m}\}}
A
=
(
λ
1
1
⋯
λ
n
1
⋮
⋱
⋮
λ
1
m
⋯
λ
n
m
)
{\displaystyle A={\begin{pmatrix}\lambda _{1}^{1}&\cdots &\lambda _{n}^{1}\\\vdots &\ddots &\vdots \\\lambda _{1}^{m}&\cdots &\lambda _{n}^{m}\end{pmatrix}}}
Aquesta matriu ens permet calcular les coordenades de la imatge d'un vector:
w
∈
E
=
∑
i
=
1
n
w
i
u
i
{\displaystyle w\in \mathbf {E} =\sum _{i=1}^{n}w_{i}u_{i}}
f
(
w
)
∈
F
=
f
(
∑
i
=
1
n
w
i
u
i
)
=
∑
i
=
1
n
w
i
(
∑
j
=
1
m
λ
i
j
v
j
)
=
∑
j
=
1
m
(
∑
i
=
1
n
λ
i
j
w
i
)
v
j
{\displaystyle \ f(w)\in F=f(\sum _{i=1}^{n}w_{i}u_{i})=\sum _{i=1}^{n}w_{i}(\sum _{j=1}^{m}\lambda _{i}^{j}v_{j})=\sum _{j=1}^{m}(\sum _{i=1}^{n}\lambda _{i}^{j}w_{i})v_{j}}
Les coordenades de
f
(
w
)
{\displaystyle \ f(w)}
{
v
1
,
…
,
v
m
}
{\displaystyle \{v_{1},\dots ,v_{m}\}}
F
{\displaystyle \mathbf {F} }
w
j
¯
=
∑
i
=
1
n
λ
i
j
w
i
,
i
=
1
,
…
,
m
{\displaystyle {\bar {w_{j}}}=\sum _{i=1}^{n}\lambda _{i}^{j}w_{i},i=1,\dots ,m}
⇒
w
¯
=
A
⋅
w
{\displaystyle \Rightarrow {\bar {w}}=A\cdot w}
modifica
Donades dues aplicacions lineals
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
g
:
F
→
G
{\displaystyle g:\mathbf {F} \rightarrow \mathbf {G} }
{
u
1
,
…
,
u
n
}
{\displaystyle \{u_{1},\dots ,u_{n}\}}
{
v
1
,
…
,
v
m
}
{\displaystyle \{v_{1},\dots ,v_{m}\}}
{
w
1
,
…
,
w
s
}
{\displaystyle \{w_{1},\dots ,w_{s}\}}
E
{\displaystyle \mathbf {E} }
F
{\displaystyle \mathbf {F} }
G
{\displaystyle \mathbf {G} }
A
{\displaystyle \ A}
B
{\displaystyle \ B}
C
=
B
⋅
A
{\displaystyle C=B\cdot A}
g
∘
f
{\displaystyle g\circ f}
f
(
u
i
)
=
∑
j
=
1
m
a
i
j
v
j
g
(
v
j
)
=
∑
k
=
1
s
b
j
k
w
k
}
⇒
g
∘
f
(
u
i
)
=
g
(
f
(
u
i
)
)
=
g
(
∑
j
=
1
m
a
i
j
v
j
)
=
∑
j
=
1
m
a
i
j
g
(
v
j
)
=
∑
j
=
1
m
a
i
j
(
∑
k
=
1
s
b
j
k
w
k
)
=
∑
k
=
1
s
(
∑
j
=
1
m
a
i
j
b
j
k
)
w
k
⇒
{\displaystyle \left.{\begin{matrix}f(u_{i})=\sum _{j=1}^{m}a_{i}^{j}v_{j}\\g(v_{j})=\sum _{k=1}^{s}b_{j}^{k}w_{k}\end{matrix}}\right\}\Rightarrow g\circ f(u_{i})=g(f(u_{i}))=g(\sum _{j=1}^{m}a_{i}^{j}v_{j})=\sum _{j=1}^{m}a_{i}^{j}g(v_{j})=\sum _{j=1}^{m}a_{i}^{j}(\sum _{k=1}^{s}b_{j}^{k}w_{k})=\sum _{k=1}^{s}(\sum _{j=1}^{m}a_{i}^{j}b_{j}^{k})w_{k}\Rightarrow }
C
i
k
=
∑
j
=
1
m
a
i
j
b
j
k
{\displaystyle C_{i}^{k}=\sum _{j=1}^{m}a_{i}^{j}b_{j}^{k}}
(
C
=
B
⋅
A
)
{\displaystyle (C=B\cdot A)}
Sigui
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
A
{\displaystyle \ A}
{
u
1
,
…
,
u
n
}
{\displaystyle \{u_{1},\dots ,u_{n}\}}
{
v
1
,
…
,
v
m
}
{\displaystyle \{v_{1},\dots ,v_{m}\}}
E
{\displaystyle \mathbf {E} }
F
{\displaystyle \mathbf {F} }
B
{\displaystyle \ B}
{
u
1
′
,
…
,
u
n
′
}
{\displaystyle \{u_{1}',\dots ,u_{n}'\}}
{
v
1
′
,
…
,
v
m
′
}
{\displaystyle \{v_{1}',\dots ,v_{m}'\}}
f
{\displaystyle {\text{f}}\;}
B
=
Q
⋅
A
⋅
P
{\displaystyle B=Q\cdot A\cdot P}
on
P
{\displaystyle \ P}
{
u
i
′
}
{\displaystyle \{u_{i}'\}\;}
{
u
i
}
{\displaystyle \{u_{i}\}\;}
Q
{\displaystyle \ Q}
{
v
j
}
{\displaystyle \{v_{j}\}\;}
{
v
j
′
}
{\displaystyle \{v_{j}'\}\;}
L'espai dual és l'espai de les aplicacions lineals que van de
E
{\displaystyle \mathbf {E} }
R
{\displaystyle \mathbb {R} }
E
→
R
{\displaystyle \mathbf {E} \rightarrow \mathbb {R} }
Les aplicacions lineals a
R
{\displaystyle \mathbb {R} }
L
(
E
,
R
)
=
E
∗
{\displaystyle {\mathcal {L}}(\mathbf {E} ,\mathbb {R} )=\mathbf {E^{*}} }
E
{\displaystyle \mathbf {E} }
L
(
E
,
R
)
{\displaystyle {\mathcal {L}}(\mathbf {E} ,\mathbb {R} )}
aplicacions lineals de
E
{\displaystyle \mathbf {E} }
R
{\displaystyle \mathbb {R} }
E
∗
{\displaystyle \mathbf {E^{*}} }
E
{\displaystyle \mathbf {E} }
E
{\displaystyle \mathbf {E} }
dim
L
(
E
,
R
)
=
dim
E
⋅
dim
R
⏟
1
=
dim
E
{\displaystyle \dim {\mathcal {L}}(\mathbf {E} ,\mathbb {R} )=\dim \mathbf {E} \cdot \underbrace {\dim \mathbb {R} } _{\text{1}}=\dim \mathbf {E} }
⇒
dim
E
∗
=
dim
E
{\displaystyle \Rightarrow \dim \mathbf {E^{*}} =\dim \mathbf {E} }
Donada una base de
E
=
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \mathbf {E} =\{u_{1},...,u_{n}\}}
u
i
′
:
{\displaystyle u_{i}':}
E
→
R
{\displaystyle \mathbf {E} \rightarrow \mathbb {R} }
u
j
↦
0
{\displaystyle u_{j}\mapsto 0}
si
j
≠
i
{\displaystyle \quad {\text{si }}~j\neq i}
u
j
↦
1
{\displaystyle u_{j}\mapsto 1}
si
j
=
i
{\displaystyle \quad {\text{si }}~j=i}
u
i
′
(
u
j
)
=
δ
i
j
=
{
1
si
i
=
j
0
si
i
≠
j
{\displaystyle u_{i}^{'}(u_{j})=\delta _{ij}=\left\{{\begin{matrix}1&{\mbox{si}}&i=j\\0&{\mbox{si}}&i\neq j\end{matrix}}\right.}
On
u
i
′
{\displaystyle u_{i}'}
u
j
{\displaystyle u_{j}}
δ
i
j
{\displaystyle \delta _{ij}}
delta de Kronecker .
Les aplicacions
{
u
i
′
}
(
i
=
1
,
.
.
.
,
n
)
{\displaystyle \{u_{i}'\}(i=1,...,n)}
E
∗
{\displaystyle \mathbf {E} ^{*}}
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \{u_{1},...,u_{n}\}}
Suposem que
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \{u_{1},...,u_{n}\}}
{
v
1
,
.
.
.
,
v
n
}
{\displaystyle \{v_{1},...,v_{n}\}}
E
{\displaystyle \mathbf {E} }
u
1
=
v
1
{\displaystyle u_{1}=v_{1}}
{
u
1
′
,
.
.
.
,
u
n
′
}
{\displaystyle \{u_{1}',...,u_{n}'\}}
{
v
1
′
,
.
.
.
,
v
n
′
}
{\displaystyle \{v_{1}',...,v_{n}'\}}
u
1
′
{\displaystyle u_{1}'}
v
1
′
{\displaystyle v_{1}'}
Sigui
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \{u_{1},...,u_{n}\}}
E
{\displaystyle \mathbf {E} }
{
u
1
′
,
.
.
.
,
u
n
′
}
{\displaystyle \{u_{1}',...,u_{n}'\}}
ω
∈
E
∗
{\displaystyle \omega \in \mathbf {E} ^{*}}
{
u
1
′
,
.
.
.
,
u
n
′
}
{\displaystyle \{u_{1}',...,u_{n}'\}}
(
ω
(
u
1
)
,
.
.
.
,
ω
(
u
n
)
)
{\displaystyle (\omega (u_{1}),...,\omega (u_{n}))}
ω
:
{\displaystyle \omega :}
E
→
R
{\displaystyle \mathbf {E} \rightarrow \mathbb {R} }
u
1
↦
ω
(
u
1
)
{\displaystyle u_{1}\mapsto \omega (u_{1})}
u
j
↦
ω
(
u
2
)
{\displaystyle u_{j}\mapsto \omega (u_{2})}
⋮
{\displaystyle \vdots }
u
n
↦
ω
(
u
n
)
{\displaystyle u_{n}\mapsto \omega (u_{n})}
ω
=
α
1
u
1
′
+
,
,
,
+
α
n
u
n
′
α
i
=
ω
(
u
i
)
i
=
1
,
.
.
.
,
n
{\displaystyle \omega =\alpha _{1}u_{1}'+,,,+\alpha _{n}u_{n}'~~~~~~~~~~\alpha _{i}=\omega (u_{i})~~~i=1,...,n}
⇒
ω
=
ω
(
u
1
)
⋅
u
1
′
+
.
.
.
+
ω
(
u
n
)
⋅
u
n
′
{\displaystyle \Rightarrow \omega =\omega (u_{1})\cdot u_{1}'+...+\omega (u_{n})\cdot u_{n}'}
ω
=
∑
i
=
1
n
ω
(
u
i
)
⋅
u
i
′
{\displaystyle \omega =\sum _{i=1}^{n}\omega (u_{i})\cdot u_{i}'}
Per tot vector
u
k
{\displaystyle u_{k}}
E
{\displaystyle \mathbf {E} }
(
∑
i
=
1
n
ω
(
u
i
)
⋅
u
i
′
)
(
u
k
)
=
∑
i
=
1
n
ω
(
u
i
)
⋅
u
i
′
(
u
k
)
=
ω
(
u
1
)
⋅
u
1
′
(
u
k
)
⏟
0
+
.
.
.
+
ω
(
u
k
)
⋅
u
k
′
(
u
k
)
⏟
1
+
.
.
.
+
ω
(
u
n
)
⋅
u
n
′
(
u
k
)
⏟
0
=
ω
(
u
k
)
{\displaystyle {\bigg (}\sum _{i=1}^{n}\omega (u_{i})\cdot u_{i}'{\bigg )}(u_{k})=\sum _{i=1}^{n}\omega (u_{i})\cdot u_{i}'(u_{k})=\omega (u_{1})\cdot \underbrace {u_{1}'(u_{k})} _{\text{0}}+...+\omega (u_{k})\cdot \underbrace {u_{k}'(u_{k})} _{\text{1}}+...+\omega (u_{n})\cdot \underbrace {u_{n}'(u_{k})} _{\text{0}}=\omega (u_{k})}
⇒
ω
=
∑
i
=
1
n
ω
(
u
i
)
⋅
u
i
′
{\displaystyle \Rightarrow \omega =\sum _{i=1}^{n}\omega (u_{i})\cdot u_{i}'}
Fixada una aplicació lineal
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
F
∗
=
L
(
F
,
R
)
{\displaystyle \mathbf {F} ^{*}={\mathcal {L}}(\mathbf {F} ,\mathbb {R} )}
ω
∈
F
∗
{\displaystyle \omega \in \mathbf {F} ^{*}}
f
{\displaystyle f}
ω
∘
f
∈
E
∗
{\displaystyle \omega \circ f\in \mathbf {E} ^{*}}
f
′
{\displaystyle f'}
f
{\displaystyle f}
f
′
:
F
∗
→
E
∗
ω
↦
ω
∘
f
{\displaystyle {\begin{matrix}f':&\mathbf {F} ^{*}\rightarrow \mathbf {E} ^{*}\\&\omega \mapsto \omega \circ f\end{matrix}}}
i té les següents propietats:
f
′
(
ω
+
v
)
=
(
ω
+
v
)
∘
f
=
(
ω
∘
f
)
+
(
v
∘
f
)
=
f
′
(
ω
)
+
f
′
(
v
)
{\displaystyle f'(\omega +v)=(\omega +v)\circ f=(\omega \circ f)+(v\circ f)=f'(\omega )+f'(v)}
f
′
(
λ
ω
)
=
(
λ
ω
)
∘
f
=
λ
(
ω
∘
f
)
=
λ
f
′
(
ω
)
{\displaystyle f'(\lambda \omega )=(\lambda \omega )\circ f=\lambda (\omega \circ f)=\lambda f'(\omega )}
(
g
∘
f
)
′
=
f
′
∘
g
′
{\displaystyle (g\circ f)'=f'\circ g'}
(
g
∘
f
)
′
(
ω
)
=
ω
∘
(
g
∘
f
)
=
(
ω
∘
g
)
∘
f
=
f
′
(
ω
∘
g
)
=
f
′
(
g
′
(
ω
)
)
=
f
′
∘
g
′
(
ω
)
{\displaystyle (g\circ f)'(\omega )=\omega \circ (g\circ f)=(\omega \circ g)\circ f=f'(\omega \circ g)=f'(g'(\omega ))=f'\circ g'(\omega )}
f
:
E
→
F
{\displaystyle f:\mathbf {E} \rightarrow \mathbf {F} }
A
=
(
a
i
j
)
{\displaystyle A=(a_{i}^{j})}
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \{u_{1},...,u_{n}\}}
{
v
1
,
.
.
.
,
v
m
}
{\displaystyle \{v_{1},...,v_{m}\}}
E
{\displaystyle \mathbf {E} }
F
{\displaystyle \mathbf {F} }
f
′
:
F
∗
→
E
∗
{\displaystyle f':\mathbf {F} ^{*}\rightarrow \mathbf {E} ^{*}}
B
=
(
b
i
j
)
{\displaystyle B=(b_{i}^{j})}
{
v
1
,
.
.
.
,
v
m
}
{\displaystyle \{v_{1},...,v_{m}\}}
{
u
1
,
.
.
.
,
u
n
}
{\displaystyle \{u_{1},...,u_{n}\}}
F
∗
{\displaystyle \mathbf {F} ^{*}}
E
∗
{\displaystyle \mathbf {E} ^{*}}
La matriu de l'aplicació dual
f
′
{\displaystyle f'}
A
{\displaystyle A}
B
=
(
b
i
j
)
=
(
a
j
i
)
=
A
t
{\displaystyle B=(b_{i}^{j})=(a_{j}^{i})=A^{t}}
b
i
j
=
(
f
′
(
v
i
′
)
)
(
u
j
)
=
(
v
i
′
∘
f
)
(
u
j
)
=
v
i
′
(
f
(
u
j
)
)
=
v
i
′
(
∑
k
=
1
m
a
j
k
v
k
)
=
∑
k
=
1
m
a
j
k
v
i
′
(
v
k
)
=
a
j
1
v
i
′
(
v
1
)
⏟
0
+
.
.
.
+
a
j
i
v
i
′
(
v
i
)
⏟
1
+
.
.
.
+
a
j
m
v
i
′
(
v
m
)
⏟
0
=
a
j
i
{\displaystyle b_{i}^{j}=(f'(v_{i}'))(u_{j})=(v_{i}'\circ f)(u_{j})=v_{i}'(f(u_{j}))=v_{i}'(\sum _{k=1}^{m}a_{j}^{k}v_{k})=\sum _{k=1}^{m}a_{j}^{k}v_{i}'(v_{k})=a_{j}^{1}\underbrace {v_{i}'(v_{1})} _{\text{0}}+...+a_{j}^{i}\underbrace {v_{i}'(v_{i})} _{\text{1}}+...+a_{j}^{m}\underbrace {v_{i}'(v_{m})} _{\text{0}}=a_{j}^{i}}
⇒
b
i
j
=
a
j
i
⇒
B
=
A
t
{\displaystyle \Rightarrow b_{i}^{j}=a_{j}^{i}\Rightarrow B=A^{t}}
