22.3: Basis Vectors

Consider the following example. We claim that the following set of vectors form a basis for \(R^3\):

\[B = \{(2,1, 3), (-1,6, 0), (3, 4, -10) \} \nonumber \]

If these vectors form a basis they must be linearly independent and Span the entire space of \(R^3\)

%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');

Do This

Create a \(3 \times 3\) numpy matrix \(A\) where the columns of \(A\) form are the basis vectors.

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.matrix(A,'68b81f1c1041158b519936cb1a2e4d6b');

Do This

Using python, calculate the determinant of matrix \(A\).

# Put your answer to the above question here. 

Do This

Using python, calculate the inverse of \(A\).

# Put your answer to the above question here.

Do This

Using python, calculate the rank of \(A\).

# Put your answer to the above question here.

Do This

Using python, calculate the reduced row echelon form of \(A\).

# Put your answer to the above question here. 

Do This

Using the above \(A\) and the vector \(b=(1,3,2)\). What is the solution to \(Ax=b\)?

#Put your answer to the above question here.

from answercheck import checkanswer

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

Turns out a matrix where column vectors are formed from basis vectors a lot of interesting properties and the following statements are equivalent.

• The column vectors of \(A\) form a basis for \(R^n\)
• \(|A| \neq 0\)
• \(A\) is invertible.
• \(A\) is row equivalent to \(I_n\) (i.e. it’s reduced row echelon form is \(I_n\))
• The system of equations \(Ax=b\) has a unique solution.
• \(rank(A) = n\)

Not all matrices follow the above statements but the ones that do are used throughout linear algebra so it is important that we know these properties.