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MATRIZ POR UN VECTOR
Si
A
m
×
n
{\displaystyle A_{m\times n}}
B
n
×
1
{\displaystyle B_{n\times 1}}
×
{\displaystyle \times }
A
b
1
,
.
.
.
.
.
.
,
a
b
n
{\displaystyle Ab_{1},......,ab_{n}}
b
1
,
.
.
.
.
.
b
n
{\displaystyle b_{1},.....b_{n}}
A
m
×
n
{\displaystyle A_{m\times n}}
B
n
×
1
{\displaystyle B_{n\times 1}}
∈
R
{\displaystyle \in \mathbb {R} }
A=
(
a
11
a
12
a
21
a
22
)
{\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}
(
b
1
b
2
)
{\displaystyle {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}
×
{\displaystyle \times }
(
a
11
a
12
a
21
a
22
)
×
(
b
1
b
2
)
{\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}\times {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}
(
a
11
b
1
+
a
12
b
2
a
21
b
1
+
a
22
b
2
)
{\displaystyle {\begin{pmatrix}a_{11}b_{1}+a_{12}b_{2}\\a_{21}b_{1}+a_{22}b_{2}\end{pmatrix}}}
Nota: Cuando multiplicamos una matriz por un vector, es necesario que el número de elementos del vector coincida con el número de columnas de la matriz. Si no es así, la multiplicación no está definida..
Para tener una visión mas general se puede escribir como:
D=
(
d
11
d
12
⋯
d
1
n
d
21
d
22
⋯
d
2
n
⋮
⋮
⋮
⋮
d
m
1
d
m
2
⋯
d
m
n
)
{\displaystyle {\begin{pmatrix}d_{11}&d_{12}&\cdots &d_{1n}\\d_{21}&d_{22}&\cdots &d_{2n}\\\vdots &\vdots &\vdots &\vdots \\d_{m1}&d_{m2}&\cdots &d_{mn}\end{pmatrix}}}
(
f
1
f
2
⋮
f
n
)
{\displaystyle {\begin{pmatrix}f_{1}\\f_{2}\\\vdots \\f_{n}\end{pmatrix}}}
×
{\displaystyle \times }
(
d
11
d
12
⋯
d
1
n
d
21
d
22
⋯
d
2
n
⋮
⋮
⋮
⋮
d
m
1
d
m
2
⋯
d
m
n
)
×
(
f
1
f
2
⋮
f
n
)
{\displaystyle {\begin{pmatrix}d_{11}&d_{12}&\cdots &d_{1n}\\d_{21}&d_{22}&\cdots &d_{2n}\\\vdots &\vdots &\vdots &\vdots \\d_{m1}&d_{m2}&\cdots &d_{mn}\end{pmatrix}}\times {\begin{pmatrix}f_{1}\\f_{2}\\\vdots \\f_{n}\end{pmatrix}}}
(
d
11
f
1
+
d
12
f
2
+
⋯
+
d
1
n
f
n
d
21
f
1
+
d
22
f
2
+
⋯
+
d
2
n
f
n
⋮
d
m
1
f
1
+
d
m
2
f
2
+
⋯
+
d
m
n
f
n
)
{\displaystyle {\begin{pmatrix}d_{11}f_{1}+d_{12}f_{2}+\cdots +d_{1n}f_{n}\\d_{21}f_{1}+d_{22}f_{2}+\cdots +d_{2n}f_{n}\\\vdots \\d_{m1}f_{1}+d_{m2}f_{2}+\cdots +d_{mn}f_{n}\end{pmatrix}}}
