how to understand empty dimension in python numpy array?

Let me just give you one example that illustrate one important difference.

d1 = np.array([1,2,3,4,5]) # array([1, 2, 3, 4, 5])d1.shape -> (5,) # row array.    d1.size -> 5# Note: d1.T is the same as d1.d2 = d1[np.newaxis] # array([[1, 2, 3, 4, 5]]). Note extra []d2.shape -> (1,5) d2.size -> 5# Note: d2.T will give a column arrayarray([[1],       [2],       [3],       [4],       [5]])d2.T.shape -> (5,1)

I also thought ndarrays would represent even 1-d arrays as 2-d arrays with a thickness of 1. Maybe because of the name "ndarray" makes us think high dimensional, however, n can be 1, so ndarrays can just have one dimension.

Compare these

x = np.array([[1], [2], [3], [4]])x.shape# (4, 1)x = np.array([[1, 2, 3, 4]])x.shape#(1, 4)x = np.array([1, 2, 3, 4])x.shape#(4,)

and (4,) means (4).

If I reshape x and back to (4), it comes back to original

x.shape = (2,2)x# array([[1, 2],#       [3, 4]])x.shape = (4)x# array([1, 2, 3, 4])

The main thing to understand here is that indexing with an integer is different than indexing with a slice. For example, when you index a 1d array or a list with an integer you get a scalar but when you index with a slice, you get an array or a list respectively. The same thing applies to 2d+ arrays. So for example:

# Make a 3d array:import numpy as nparray = np.arange(60).reshape((3, 4, 5))# Indexing with ints gives a scalarprint array[2, 3, 4] == 59# True# Indexing with slices gives a 3d arrayprint array[:2, :2, :2].shape# (2, 2, 2)# Indexing with a mix of slices and ints will give an array with < 3 dimsprint array[0, :2, :3].shape# (2, 3)print array[:, 2, 0:1].shape# (3, 1)

This can be really useful conceptually, because sometimes its great to think of an array as a collection of vectors, for example I can represent N points in space as an (N, 3) array:

n_points = np.random.random([10, 3])point_2 = n_points[2]print all(point_2 == n_points[2, :])# True