Zero-based indexing

In math we use 1-based indexing, but in many programming languages including Python, we use 0-based indexing.

For example, a three dimensional vector $\vec{v} \in \mathbb{R}^3$ are denoted as $(v_1,v_2,v_3)$ in components form.

In Python a vector vec can be denoted as a list (or a Matrix), and its three components will be vec[0], vec[1], and vec[2]. The variable we to index the components is assumed to start at zero, not one.



In [1]:

    
from sympy import Matrix, symbols
a, b, c = symbols('a b c')

vec = Matrix([a,b,c])
vec









    Out[1]:





Matrix([
[a],
[b],
[c]])



In [2]:

    
vec[0]









    Out[2]:





a



In [3]:

    
vec[1]









    Out[3]:





b



In [4]:

    
vec[2]









    Out[4]:





c



In [5]:

    
# trying to access vec[3] leads to an error, 
# since a 3 in 0-based indexing is accessing a fourth component which doesn't exist
vec[3]









    



---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-5-ad6368f205ef> in <module>()
      1 # trying to access vec[3] leads to an error,
      2 # since a 3 in 0-based indexing is accessing a fourth component which doesn't exist
----> 3 vec[3]

/Users/ivan/Projects/Minireference/content/noBSLAnotebooks/pydev/lib/python3.5/site-packages/sympy/matrices/dense.py in __getitem__(self, key)
     94             if isinstance(key, slice):
     95                 return self._mat[key]
---> 96             return self._mat[a2idx(key)]
     97 
     98     def __setitem__(self, key, value):

IndexError: list index out of range

Ranges



In [6]:

    
# a list of five numbers
list(range(0,5))
# Note the second argument of the function range specifies
#      the first number not to be included in the range...









    Out[6]:





[0, 1, 2, 3, 4]



In [ ]:

SymPy shorthand for ranges



In [7]:

    
# consider a 2x3 matrix
A = Matrix([
        [1,2,3],
        [4,5,6]])
A









    Out[7]:





Matrix([
[1, 2, 3],
[4, 5, 6]])



In [14]:

    
# can access individual entry on row i, col j using A[i-1,j-1]
print('top left =',    A[0,0], '    top right =', A[0,2])
print('bottom left =', A[1,0], ' bottom right =', A[1,2])









    



top left = 1     top right = 3
bottom left = 4  bottom right = 6



In [9]:

    
# can access entire colum j using A[:,j-1]
A[:,0]









    Out[9]:





Matrix([
[1],
[4]])



In [ ]:

    
# can access entire row i using A[i-1,:]
A[0,:]



In [ ]:

    
# can access left 2x2 submatrix using i:j shorhand range notation
A[0:2,0:2]



In [ ]:

    
# the above notation is MATLAB-like syntax sugar that translates to
A[range(0,2),range(0,2)]



In [ ]: