Un espacio vectorial
V
{\displaystyle \mathbb {V} }
cuerpo
K
{\displaystyle \mathbb {K} }
⊕
:
V
×
V
→
V
{\displaystyle \oplus :\mathbb {V} \times \mathbb {V} \rightarrow \mathbb {V} }
(
u
,
v
)
→
u
⊕
v
{\displaystyle (u,v)\rightarrow u\oplus v}
⊕
{\displaystyle \oplus }
V
{\displaystyle \mathbb {V} }
⊗
:
K
×
V
→
V
{\displaystyle \otimes :\mathbb {K} \times \mathbb {V} \rightarrow \mathbb {V} }
(
α
,
u
)
→
α
⊗
u
{\displaystyle (\alpha ,u)\rightarrow \alpha \otimes u}
⊗
{\displaystyle \otimes }
K
{\displaystyle \mathbb {K} }
V
{\displaystyle \mathbb {V} }
u
⊕
v
=
v
⊕
u
{\displaystyle u\oplus v=v\oplus u}
∀
u
,
v
∈
V
{\displaystyle \forall u,v\in \mathbb {V} }
u
⊕
(
v
⊕
w
)
=
(
u
⊕
v
)
⊕
w
{\displaystyle u\oplus (v\oplus w)=(u\oplus v)\oplus w}
∀
u
,
v
,
w
∈
V
{\displaystyle \forall u,v,w\in \mathbb {V} }
Existencia de elemento neutro ante
⊕
{\displaystyle \oplus }
∀
u
∈
V
{\displaystyle \forall u\in \mathbb {V} }
∃
e
∈
V
{\displaystyle \exists e\in \mathbb {V} }
∋
{\displaystyle \ni }
u
⊕
e
=
u
{\displaystyle u\oplus e=u}
Existencia de elemento opuesto ante
⊕
{\displaystyle \oplus }
∀
u
∈
V
{\displaystyle \forall u\in \mathbb {V} }
∃
u
′
{\displaystyle \exists u\prime }
∋
{\displaystyle \ni }
u
⊕
u
′
=
e
{\displaystyle u\oplus u\prime =e}
(
α
∙
β
)
⊗
u
=
α
⊗
(
β
⊗
u
)
{\displaystyle (\alpha \bullet \beta )\otimes u=\alpha \otimes (\beta \otimes u)}
∀
α
,
β
∈
K
{\displaystyle \forall \alpha ,\beta \in \mathbb {K} }
∀
u
∈
V
{\displaystyle \forall u\in \mathbb {V} }
Propiedad distributiva para la opearación (+) entre escalares
(
α
(
+
)
β
)
⊗
u
=
α
⊗
u
⊕
β
⊗
u
{\displaystyle (\alpha (+)\beta )\otimes u=\alpha \otimes u\oplus \beta \otimes u}
∀
α
,
β
∈
K
{\displaystyle \forall \alpha ,\beta \in \mathbb {K} }
∀
u
∈
V
{\displaystyle \forall u\in \mathbb {V} }
Propiedad distributiva para la operación
⊕
{\displaystyle \oplus }
V
{\displaystyle \mathbb {V} }
α
⊗
(
u
⊕
v
)
=
α
⊗
u
⊕
α
⊗
v
{\displaystyle \alpha \otimes (u\oplus v)=\alpha \otimes u\oplus \alpha \otimes v}
∀
α
∈
K
{\displaystyle \forall \alpha \in \mathbb {K} }
∀
u
,
v
∈
V
{\displaystyle \forall u,v\in \mathbb {V} }
Existencia de elemento neutro ante la operación
⊗
{\displaystyle \otimes }
∀
u
∈
V
{\displaystyle \forall u\in \mathbb {V} }
∃
ϵ
∈
K
{\displaystyle \exists \epsilon \in \mathbb {K} }
∋
{\displaystyle \ni }
ϵ
⊗
u
=
u
{\displaystyle \epsilon \otimes u=u}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
{\displaystyle \mathbb {R} }
En efecto:
⊕
:
R
n
×
R
n
→
R
n
{\displaystyle \oplus :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}
(
a
,
b
)
→
a
+
b
{\displaystyle (a,b)\rightarrow a+b}
∗
:
R
×
R
n
→
R
n
{\displaystyle *:\mathbb {R} \times \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}
(
α
,
a
)
→
α
∗
a
{\displaystyle (\alpha ,a)\rightarrow \alpha *a}
Ante la suma
∀
a
,
b
∈
R
n
{\displaystyle \forall a,b\in \mathbb {R} ^{n}}
a
⊕
b
=
b
⊕
a
{\displaystyle a\oplus b=b\oplus a}
∀
a
,
b
,
c
∈
R
n
{\displaystyle \forall a,b,c\in \mathbb {R} ^{n}}
a
⊕
(
b
⊕
c
)
=
(
a
⊕
b
)
⊕
c
{\displaystyle a\oplus (b\oplus c)=(a\oplus b)\oplus c}
∀
a
∈
R
n
∃
0
∈
R
n
{\displaystyle \forall a\in \mathbb {R} ^{n}\exists 0\in \mathbb {R} ^{n}}
a
⊕
0
=
a
{\displaystyle a\oplus 0=a}
∀
a
∈
R
n
∃
(
−
1
)
a
∈
R
n
{\displaystyle \forall a\in \mathbb {R} ^{n}\exists (-1)a\in \mathbb {R} ^{n}}
a
⊕
(
−
1
)
a
=
0
{\displaystyle a\oplus (-1)a=0}
Ante el producto por escalares
∀
α
,
β
∈
R
,
∀
a
∈
R
n
{\displaystyle \forall \alpha ,\beta \in \mathbb {R} ,\forall a\in \mathbb {R} ^{n}}
(
α
⋅
β
)
∗
a
=
α
∗
(
β
∗
a
)
{\displaystyle (\alpha \cdot \beta )*a=\alpha *(\beta *a)}
∀
α
,
β
∈
R
,
∀
a
∈
R
n
{\displaystyle \forall \alpha ,\beta \in \mathbb {R} ,\forall a\in \mathbb {R} ^{n}}
(
α
⋅
β
)
∗
a
=
α
∗
a
+
β
∗
a
{\displaystyle (\alpha \cdot \beta )*a=\alpha *a+\beta *a}
∀
α
∈
R
,
∀
a
,
b
∈
R
n
{\displaystyle \forall \alpha \in \mathbb {R} ,\forall a,b\in \mathbb {R} ^{n}}
α
∗
(
a
⊕
b
)
=
α
∗
a
⊕
α
∗
b
{\displaystyle \alpha *(a\oplus b)=\alpha *a\oplus \alpha *b}
∀
a
∈
R
n
,
∃
α
∈
R
{\displaystyle \forall a\in \mathbb {R} ^{n},\exists \alpha \in \mathbb {R} }
α
∗
a
=
a
{\displaystyle \alpha *a=a}
2.El conjunto de todos los polinomios
3.El conjunto de todas las funciones continuas
Una matriz es un ordenamiento de elementos de un cuerpo, representado por filas y columnas, por ejemplo:
A
=
(
a
11
a
12
…
a
1
n
a
21
a
22
…
a
2
n
⋮
⋮
⋮
⋮
a
i
1
a
i
2
…
a
i
n
⋮
⋮
⋮
⋮
a
m
1
a
m
2
…
a
m
n
)
,
∋
A
∈
M
m
,
n
(
K
)
{\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}&a_{i2}&\ldots &a_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}},\ni A\in M_{m,n}(\mathbb {K} )}
Donde:
M
m
,
n
(
K
)
{\displaystyle M_{m,n}(\mathbb {K} )}
m
−
f
i
l
a
s
,
n
−
c
o
l
u
m
n
a
s
{\displaystyle m-filas,n-columnas}
K
{\displaystyle \mathbb {K} }
Normalmente la
i
,
j
{\displaystyle i,j}
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (a_{ij})}
Si m=n la matriz se suele llamar cuadrada, por ejemplo:
C
=
(
a
11
a
12
…
a
1
n
a
21
a
22
…
a
2
n
⋮
⋮
⋮
⋮
a
j
1
a
j
2
…
a
j
n
⋮
⋮
⋮
⋮
a
n
1
a
n
2
…
a
n
n
)
,
∋
C
∈
M
m
,
n
(
K
)
{\displaystyle C={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{j1}&a_{j2}&\ldots &a_{jn}\\\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{pmatrix}},\ni C\in M_{m,n}(\mathbb {K} )}
Que tiene por entrada
(
c
i
j
)
{\displaystyle (c_{ij})}
∀
i
,
j
∈
1
,
.
.
.
,
n
{\displaystyle \forall i,j\in {1,...,n}}
En particular:
B
=
(
3
5
20
…
2
20
−
30
…
π
⋮
⋮
⋮
⋮
e
34
×
10
−
23
…
590
⋮
⋮
⋮
⋮
log
12
20
897
…
4576
)
,
∋
B
∈
M
n
,
n
(
R
)
{\displaystyle B={\begin{pmatrix}3&5^{20}&\ldots &{\sqrt {2}}\\20&-30&\ldots &\pi \\\vdots &\vdots &\vdots &\vdots \\e&34\times 10^{-23}&\ldots &590\\\vdots &\vdots &\vdots &\vdots \\\log _{12}20&897&\ldots &4576\end{pmatrix}},\ni B\in M_{n,n}(\mathbb {R} )}
Si
m
>
n
{\displaystyle m>n}
D
=
(
a
11
a
21
⋮
a
i
1
⋮
a
m
1
)
,
∋
D
∈
M
m
,
1
(
K
)
{\displaystyle D={\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{i1}\\\vdots \\a_{m1}\end{pmatrix}},\ni D\in M_{m,1}(\mathbb {K} )}
que en particular puede representar un vector en
R
m
{\displaystyle \mathbb {R} ^{m}}
M
m
,
1
(
R
)
{\displaystyle M_{m,1}(\mathbb {R} )}
Como podemos observar la condensación de notación en forma matricial es una ventaja imprescindible, pues, al trabajar con grandes cantidades numéricas se ahorra memoria al igual que trabajo en sí mismo.
Ahora vamos a exponer un método para operar con matrices.
Definición .
Sean
A
{\displaystyle \mathrm {A} }
B
{\displaystyle \mathrm {B} }
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
(
b
i
j
)
{\displaystyle (\mathrm {b_{ij}} )}
A
{\displaystyle \mathrm {A} }
igual a
B
{\displaystyle \mathrm {B} }
(
a
i
j
)
=
(
b
i
j
)
{\displaystyle (\mathrm {a_{ij})=(b_{ij}} )}
Definición .
Sean
A
{\displaystyle \mathrm {A} }
B
{\displaystyle \mathrm {B} }
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
la adición de
A
{\displaystyle \mathrm {A} }
B
{\displaystyle \mathrm {B} }
A
+
B
=
(
(
a
i
j
)
+
(
b
i
j
)
)
{\displaystyle \mathrm {A+B=((a_{ij})+(b_{ij})} )}
Observación:
notar que la adición se genera entrada con entrada.
Ejemplo :
A
=
(
a
11
a
12
…
a
1
n
a
21
a
22
…
a
2
n
⋮
⋮
⋮
⋮
a
i
1
a
i
2
…
a
i
n
⋮
⋮
⋮
⋮
a
m
1
a
m
2
…
a
m
n
)
,
B
=
(
b
11
b
12
…
b
1
n
b
21
b
22
…
b
2
n
⋮
⋮
⋮
⋮
b
i
1
b
i
2
…
b
i
n
⋮
⋮
⋮
⋮
b
m
1
b
m
2
…
b
m
n
)
,
∋
A
,
B
∈
M
m
,
n
(
K
)
{\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}&a_{i2}&\ldots &a_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}},B={\begin{pmatrix}b_{11}&b_{12}&\ldots &b_{1n}\\b_{21}&b_{22}&\ldots &b_{2n}\\\vdots &\vdots &\vdots &\vdots \\b_{i1}&b_{i2}&\ldots &b_{in}\\\vdots &\vdots &\vdots &\vdots \\b_{m1}&b_{m2}&\ldots &b_{mn}\end{pmatrix}},\ni A,B\in M_{m,n}(\mathbb {K} )}
A
+
B
=
(
a
11
+
b
11
a
12
+
b
12
…
a
1
n
+
b
1
n
a
21
+
b
21
a
22
+
b
22
…
a
2
n
+
b
2
n
⋮
⋮
⋮
⋮
a
i
1
+
b
i
1
a
i
2
+
b
i
2
…
a
i
n
+
b
i
n
⋮
⋮
⋮
⋮
a
m
1
+
b
m
1
a
m
2
+
b
m
2
…
a
m
n
+
b
m
n
)
,
∋
[
A
+
B
]
∈
M
m
,
n
(
K
)
{\displaystyle A+B={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\ldots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\ldots &a_{2n}+b_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}+b_{i1}&a_{i2}+b_{i2}&\ldots &a_{in}+b_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\ldots &a_{mn}+b_{mn}\end{pmatrix}},\ni [A+B]\in M_{m,n}(\mathbb {K} )}
Definición:
Considere
A
{\displaystyle \mathrm {A} }
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
α
{\displaystyle \alpha }
∈
K
{\displaystyle \in \mathbb {K} }
α
⋅
A
{\displaystyle \alpha \cdot A}
α
⋅
A
=
(
α
⋅
a
11
α
⋅
a
12
…
α
⋅
a
1
n
α
⋅
a
21
α
⋅
a
22
…
α
⋅
a
2
n
⋮
⋮
⋮
⋮
α
⋅
a
i
1
α
⋅
a
i
2
…
α
⋅
a
i
n
⋮
⋮
⋮
⋮
α
⋅
a
m
1
α
⋅
a
m
2
…
α
⋅
a
m
n
)
,
∋
[
α
⋅
A
]
∈
M
m
,
n
(
K
)
{\displaystyle \alpha \cdot A={\begin{pmatrix}\alpha \cdot a_{11}&\alpha \cdot a_{12}&\ldots &\alpha \cdot a_{1n}\\\alpha \cdot a_{21}&\alpha \cdot a_{22}&\ldots &\alpha \cdot a_{2n}\\\vdots &\vdots &\vdots &\vdots \\\alpha \cdot a_{i1}&\alpha \cdot a_{i2}&\ldots &\alpha \cdot a_{in}\\\vdots &\vdots &\vdots &\vdots \\\alpha \cdot a_{m1}&\alpha \cdot a_{m2}&\ldots &\alpha \cdot a_{mn}\end{pmatrix}},\ni [\alpha \cdot A]\in M_{m,n}(\mathbb {K} )}
Observación :
En las dos definiciones anteriores la cerradura se debe a que son elementos de un cuerpo, el cual, posee esta propiedad.
Sin mucha dificultad se puede demostrar que el conjunto
M
m
,
n
(
K
)
{\displaystyle M_{m,n}(\mathbb {K} )}
K
{\displaystyle \mathbb {K} }
Definición:
Sea
A
{\displaystyle \mathrm {A} }
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
B
{\displaystyle \mathrm {B} }
n
×
q
{\displaystyle n\times q}
(
b
i
j
)
{\displaystyle (\mathrm {b_{ij}} )}
A
∙
B
{\displaystyle A\bullet B}
A
{\displaystyle \mathrm {A} }
B
{\displaystyle \mathrm {B} }
A
∙
B
=
(
∑
j
=
i
n
a
1
j
b
j
1
∑
j
=
1
n
a
1
j
b
j
2
…
∑
j
=
1
n
a
1
j
b
j
q
∑
j
=
1
n
a
2
j
b
j
1
∑
j
=
1
n
a
2
j
b
j
2
…
∑
j
=
1
n
a
2
j
b
j
q
⋮
⋮
⋮
⋮
∑
j
=
1
n
a
i
j
b
j
1
∑
j
=
1
n
a
i
j
b
j
2
…
∑
j
=
1
n
a
i
j
b
j
q
⋮
⋮
⋮
⋮
∑
j
=
1
n
a
m
j
b
j
1
∑
j
=
1
n
a
m
j
b
j
2
…
∑
j
=
1
n
a
m
j
b
j
q
)
{\displaystyle A\bullet B={\begin{pmatrix}\sum _{j=i}^{n}a_{1j}b_{j1}&\sum _{j=1}^{n}a_{1j}b_{j2}&\ldots &\sum _{j=1}^{n}a_{1j}b_{jq}\\\sum _{j=1}^{n}a_{2j}b_{j1}&\sum _{j=1}^{n}a_{2j}b_{j2}&\ldots &\sum _{j=1}^{n}a_{2j}b_{jq}\\\vdots &\vdots &\vdots &\vdots \\\sum _{j=1}^{n}a_{ij}b_{j1}&\sum _{j=1}^{n}a_{ij}b_{j2}&\ldots &\sum _{j=1}^{n}a_{ij}b_{jq}\\\vdots &\vdots &\vdots &\vdots \\\sum _{j=1}^{n}a_{mj}b_{j1}&\sum _{j=1}^{n}a_{mj}b_{j2}&\ldots &\sum _{j=1}^{n}a_{mj}b_{jq}\end{pmatrix}}}
generando una matriz
C
=
A
∙
B
{\displaystyle C=A\bullet B}
m
×
q
{\displaystyle m\times q}
(
c
i
k
)
=
(
∑
j
=
1
n
a
i
j
b
j
k
)
{\displaystyle (c_{ik})=(\sum _{j=1}^{n}a_{ij}b{jk})}
Observación :
Es importante que las filas coincidan con las columnas o viceversa en la multiplicación de matrices ya que si no fuese así no se puede definir ninguna multiplicación entre estas.
Propiedades inmediatas:
Sea
A
{\displaystyle \mathrm {A} }
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
B
{\displaystyle \mathrm {B} }
n
×
q
{\displaystyle n\times q}
(
b
j
k
)
{\displaystyle (\mathrm {b_{jk}} )}
α
l
∈
K
{\displaystyle \alpha _{l}\in \mathbb {K} }
1.
(
∑
l
=
1
r
α
l
)
⋅
(
A
∙
B
)
=
(
∑
l
=
1
r
α
l
⋅
A
)
∙
B
=
A
∙
(
∑
l
=
1
r
α
l
⋅
B
)
=
(
A
∙
B
)
⋅
∑
l
=
1
r
α
l
{\displaystyle (\sum _{l=1}^{r}\alpha _{l})\cdot (A\bullet B)=(\sum _{l=1}^{r}\alpha _{l}\cdot A)\bullet B=A\bullet (\sum _{l=1}^{r}\alpha _{l}\cdot B)=(A\bullet B)\cdot \sum _{l=1}^{r}\alpha _{l}}
2.
∏
l
=
1
s
α
l
⋅
(
A
∙
B
)
=
(
∏
l
=
1
s
α
l
⋅
A
)
∙
B
=
A
∙
(
B
⋅
∏
l
=
1
s
α
l
)
=
(
A
∙
B
)
⋅
∏
l
=
1
s
α
l
{\displaystyle \prod _{l=1}^{s}\alpha _{l}\cdot (A\bullet B)=(\prod _{l=1}^{s}\alpha _{l}\cdot A)\bullet B=A\bullet (B\cdot \prod _{l=1}^{s}\alpha _{l})=(A\bullet B)\cdot \prod _{l=1}^{s}\alpha _{l}}
Definición.
Sea
A
{\displaystyle \mathrm {A} }
m
×
n
{\displaystyle m\times n}
(
a
i
j
)
{\displaystyle (\mathrm {a_{ij}} )}
la traspuesta de
A
{\displaystyle \mathrm {A} }
n
×
m
{\displaystyle n\times m}
A
T
{\displaystyle \mathrm {A} ^{T}}
(
a
j
i
)
{\displaystyle (a_{ji})}
Ejemplo:
A
=
(
b
11
b
12
…
b
1
n
b
21
b
22
…
b
2
n
⋮
⋮
⋮
⋮
b
i
1
b
i
2
…
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{\displaystyle A={\begin{pmatrix}b_{11}&b_{12}&\ldots &b_{1n}\\b_{21}&b_{22}&\ldots &b_{2n}\\\vdots &\vdots &\vdots &\vdots \\b_{i1}&b_{i2}&\ldots &b_{in}\\\vdots &\vdots &\vdots &\vdots \\b_{m1}&b_{m2}&\ldots &b_{mn}\end{pmatrix}},A^{T}={\begin{pmatrix}b_{11}&b_{12}&\ldots &b_{1m}\\b_{21}&b_{22}&\ldots &b_{2m}\\\vdots &\vdots &\vdots &\vdots \\b_{j1}&b_{i2}&\ldots &b_{jm}\\\vdots &\vdots &\vdots &\vdots \\b_{n1}&b_{n2}&\ldots &b_{nm}\end{pmatrix}}}
Propiedades:
1.
∀
A
∈
M
m
,
n
(
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{\displaystyle \forall A\in M_{m,n}(\mathbb {K} )}
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=
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{\displaystyle (\mathbf {A^{T})^{T}=A} }
2.
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∑
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l
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∑
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1
l
A
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{\displaystyle (\mathbf {\sum _{i=1}^{l}A_{i})^{T}=\sum _{i=1}^{l}A_{i}^{T}} }
3.
∀
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j
∈
K
{\displaystyle \forall \mathbf {\alpha _{j}} \in \mathbb {K} }
∀
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i
∈
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{\displaystyle \forall \mathbf {A_{i}} \in M_{m,n}(\mathbb {K} )}
∏
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{\displaystyle \mathbf {\prod _{v=1}^{s}\alpha _{v}\cdot (\sum _{i=1}^{p}A_{i})^{T}=(\prod _{v=1}^{s}\alpha _{v}\cdot \sum _{i=1}^{p}A_{i})^{T}=\sum _{i=1}^{p}A_{i}^{T}\cdot \prod _{v=1}^{s}\alpha _{v}} }
4.
∀
A
i
{\displaystyle \forall \mathbf {A_{i}} }
B
i
{\displaystyle \mathbf {B_{i}} }
∈
M
m
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{\displaystyle \in M_{m,n}(\mathbb {K} )}
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{\displaystyle \mathbf {(\sum _{i=1}^{s}A_{i}+B_{i})^{T}=\sum _{i=1}^{n}(B_{i})^{T}+\sum _{i=1}^{n}(A_{i})^{T}} }
Todo sistema de m-ecuaciones lineales con n-incógnitas (
m
×
n
{\displaystyle m\times n}
(
K
)
{\displaystyle (\mathbb {K} )}
Demostración:
Básicamente lo que necesitamos recordar es cómo se representa a tal sistema, por ejemplo se tiene:
a
11
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a
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⋯
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a
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=
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m
{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{i1}x_{1}+a_{i2}x_{2}+a_{i3}x_{3}+\cdots +a_{in}x_{n}=b_{i}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+a_{m3}x_{3}+\cdots +a_{mn}x_{n}=b_{m}\end{matrix}}}
Observemos cuidadosamente que los coeficientes
a
i
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∀
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,
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∈
1
,
2
,
.
.
.
,
n
{\displaystyle a_{ij}\forall i,j\in {1,2,...,n}}
(
a
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j
)
{\displaystyle (a_{ij})}
m
×
n
{\displaystyle m\times n}
A
=
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,
∀
A
∈
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m
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n
(
K
)
{\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}&a_{i2}&\ldots &a_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}},\forall A\in M_{m,n}(\mathbb {K} )}
Notemos que la primera ecuación es de la forma y posee la característica de la primera entrada de la multiplicación de A con una matriz
X
{\displaystyle X}
1
×
n
{\displaystyle 1\times n}
(
x
j
)
∀
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∈
1
,
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,
.
.
.
,
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{\displaystyle (\mathrm {x_{j}} )\forall j\in {1,2,...,n}}
A
∙
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=
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{\displaystyle A\bullet X={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}&a_{i2}&\ldots &a_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}\bullet {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\\\end{pmatrix}}}
Que esto evidentemente tiene que ser igual a una matriz
m
×
1
{\displaystyle m\times 1}
(
b
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{\displaystyle (b_{ij})}
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{\displaystyle A\bullet X={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}&a_{i2}&\ldots &a_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\\\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \\b_{m}\\\end{pmatrix}}=B}
![{\displaystyle A+B={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\ldots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\ldots &a_{2n}+b_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{i1}+b_{i1}&a_{i2}+b_{i2}&\ldots &a_{in}+b_{in}\\\vdots &\vdots &\vdots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\ldots &a_{mn}+b_{mn}\end{pmatrix}},\ni [A+B]\in M_{m,n}(\mathbb {K} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff611d3d824d26f1e92e9886c80fa71c61719e1)
![{\displaystyle \alpha \cdot A={\begin{pmatrix}\alpha \cdot a_{11}&\alpha \cdot a_{12}&\ldots &\alpha \cdot a_{1n}\\\alpha \cdot a_{21}&\alpha \cdot a_{22}&\ldots &\alpha \cdot a_{2n}\\\vdots &\vdots &\vdots &\vdots \\\alpha \cdot a_{i1}&\alpha \cdot a_{i2}&\ldots &\alpha \cdot a_{in}\\\vdots &\vdots &\vdots &\vdots \\\alpha \cdot a_{m1}&\alpha \cdot a_{m2}&\ldots &\alpha \cdot a_{mn}\end{pmatrix}},\ni [\alpha \cdot A]\in M_{m,n}(\mathbb {K} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c33436740dd04b1be6301fae34706e46a0a745c)
