numpy array that is (n,1) and (n,)

Question

What is the difference between a numpy array (lets say X) that has a shape of (N,1) and (N,). Aren't both of them Nx1 matrices ? The reason I ask is because sometimes computations return either one or the other.

Answer 1

This is a 1D array:

>>> np.array([1, 2, 3]).shape
(3,)

This array is a 2D but there is only one element in the first dimension:

>>> np.array([[1, 2, 3]]).shape
(1, 3)

Transposing gives the shape you are asking for:

>>> np.array([[1, 2, 3]]).T.shape
(3, 1)

Now, look at the array. Only the first column of this 2D array is filled.

>>> np.array([[1, 2, 3]]).T
array([[1],
       [2],
       [3]])

Given these two arrays:

>>> a = np.array([[1, 2, 3]])
>>> b = np.array([[1, 2, 3]]).T
>>> a
array([[1, 2, 3]])
>>> b
array([[1],
       [2],
       [3]])

You can take advantage of broadcasting:

>>> a * b
array([[1, 2, 3],
       [2, 4, 6],
       [3, 6, 9]])

The missing numbers are filled in. Think for rows and columns in table or spreadsheet.

>>> a + b
array([[2, 3, 4],
       [3, 4, 5],
       [4, 5, 6]]) 

Doing this with higher dimensions gets tougher on your imagination.