# Álgebra Lineal/Matriz por vector

MATRIZ POR UN VECTOR

Si ${\displaystyle A_{m\times n}}$ y ${\displaystyle B_{n\times 1}}$. El producto AB es la matriz m${\displaystyle \times }$1 cuyos columnas son ${\displaystyle Ab_{1},......,ab_{n}}$ . En la que ${\displaystyle b_{1},.....b_{n}}$. son columnas de B.

${\displaystyle A_{m\times n}}$ y ${\displaystyle B_{n\times 1}}$ ${\displaystyle \in \mathbb {R} }$
A=${\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}$ , B=${\displaystyle {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}$  A ${\displaystyle \times }$ B= ${\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}\times {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}a_{11}b_{1}+a_{12}b_{2}\\a_{21}b_{1}+a_{22}b_{2}\end{pmatrix}}}$

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= ${\displaystyle {\begin{pmatrix}d_{11}&d_{12}&\cdots &d_{1n}\\d_{21}&d_{22}&\cdots &d_{2n}\\\vdots &\vdots &\vdots &\vdots \\d_{m1}&d_{m2}&\cdots &d_{mn}\end{pmatrix}}}$ , F= ${\displaystyle {\begin{pmatrix}f_{1}\\f_{2}\\\vdots \\f_{n}\end{pmatrix}}}$
D ${\displaystyle \times }$ F= ${\displaystyle {\begin{pmatrix}d_{11}&d_{12}&\cdots &d_{1n}\\d_{21}&d_{22}&\cdots &d_{2n}\\\vdots &\vdots &\vdots &\vdots \\d_{m1}&d_{m2}&\cdots &d_{mn}\end{pmatrix}}\times {\begin{pmatrix}f_{1}\\f_{2}\\\vdots \\f_{n}\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}d_{11}f_{1}+d_{12}f_{2}+\cdots +d_{1n}f_{n}\\d_{21}f_{1}+d_{22}f_{2}+\cdots +d_{2n}f_{n}\\\vdots \\d_{m1}f_{1}+d_{m2}f_{2}+\cdots +d_{mn}f_{n}\end{pmatrix}}}$