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22.3: Vectores de base

( \newcommand{\kernel}{\mathrm{null}\,}\)

Considera el siguiente ejemplo. Afirmamos que el siguiente conjunto de vectores forman una base paraR3:

B={(2,1,3),(1,6,0),(3,4,10)}

Si estos vectores forman una base, deben ser linealmente independientes y abarcan todo el espacio deR3

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%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
from urllib.request import urlretrieve
sym.init_printing(use_unicode=True)
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
from urllib.request import urlretrieve
sym.init_printing(use_unicode=True)
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');
Hacer esto

Crea una matriz3×3 numpyA donde las columnas deA form sean los vectores base.

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from answercheck import checkanswer

checkanswer.matrix(A,'68b81f1c1041158b519936cb1a2e4d6b');
from answercheck import checkanswer

checkanswer.matrix(A,'68b81f1c1041158b519936cb1a2e4d6b');
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Usando python, calcula el determinante de la matrizA.

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Usando python, calcule la inversa deA.

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Usando python, calcule el rango deA.

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Usando python, calcule la forma de escalón de fila reducida deA.

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Usando lo anteriorA y el vectorb=(1,3,2). ¿Cuál es la soluciónAx=b?

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from answercheck import checkanswer

checkanswer.matrix(x,'8b0938260dfeaafc9f8e9fec0bc72f17');
from answercheck import checkanswer

checkanswer.matrix(x,'8b0938260dfeaafc9f8e9fec0bc72f17');

Resulta una matriz donde los vectores de columna se forman a partir de vectores base muchas propiedades interesantes y las siguientes declaraciones son equivalentes.

  • Los vectores de columnaA forman una base paraRn
  • |A|0
  • Aes invertible.
  • Aes fila equivalente aIn (es decir, su forma de escalón de fila reducida esIn)
  • El sistema de ecuacionesAx=b tiene una solución única.
  • rank(A)=n

No todas las matrices siguen las declaraciones anteriores sino las que sí se utilizan a lo largo del álgebra lineal por lo que es importante que conozcamos estas propiedades.


This page titled 22.3: Vectores de base is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform.

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