How to visualize/connect vectors, matrices and representations in Python and numpy arrays?

Question

I am having trouble visualizing scalars, vectors and matrices as how they are written in a math/physics class to how they would be represented in plain Python and numpy and their corresponding notions of dimensions, axes and shapes.

1. If I have a scalar, say 5

>>> b = np.array(5)
>>> np.ndim(b)
0

I have 0 dimensions for 5 but what are the axes here? There are functions for ndim and shape but not axes.

2. For a vector like this:

we say that we have 2 dimensions in physics/math class because it represents a 2D vector but it looks like numpy uses a different notion of this.

Why is it that ndim gives 1 and shape gives what the dimension is?

>>> c = np.array([1,-3])
>>> c
array([ 1, -3])
>>> c.ndim
1
>>> c.shape
(2,)

np.ndim gives 1 then?

I have looked at this tutorial on axes but haven't been able to get how the axes then apply here.

How would you represent the vector above in Python and numpy ? Would this be [1, -3] in Python or [[1], [-3]] ? How about in numpy ? Would it be np.array([1, -3]) or np.array([[1], [-3]]) ? Which I tend to write, for my eyes' sake, as

np.array([
    [1], 
    [-3]
])

3. Other than vectors, how would this matrix be represented in both plain Python and numpy ? The documentation states that we need to use np.array s instead.

It is no longer recommended to use this class, even for linear algebra. Instead use regular arrays. The class may be removed in the future.

4. When it comes to multidimensional arrays, how would I represent/visualize multiple values for all the points in a 3D cube? Say we have a Rubik's cube and each of the sub cubes has a temperature and a color represented with red, green and blue so 4 values for each cube?

Answer 1

A scalar is not an array, so it has 0 dimensions.

np.array([1,-3]) is a 1D array, so c.shape returns a tuple with only one element (2,) , just the first dimension and it's telling you there is only 1 dimension and 2 elements in that dimension.

You are correct np.array([[1], [-3]]) is the vector you have in 2. c.shape gives (2,1) meaning there are 2 rows and 1 column. c.ndim gives 2 since there are 2 dimensions x and y. It's a 2D/planar array

For 3., you would create it as np.array([[1,2,3], [4,5,6], [7,8,9]]) . shape returns (3,3) meaning 3 rows and 3 columns. ndim returns 2 because it's still a 2D/planar array.

Answer 2

A ndarray has a shape , a tuple. ndim is the length of that tuple, and may be 0. The array has ndim axes (sometimes called dimensions).

 np.array(5)

has shape () , 0 ndim and no axes.

np.array([1,2,3,4])

has (4,) shape, and 1 axis. It can be reshaped to (4,1), or (1,4) or (2,2) or even (2,1,2) or (1,4,1).

Your A can be created with

A = np.arange(1,10).reshape(3,3)

That's a 9 element 1d array reshaped to (3,3)

numpy arrays have a print display, with [] marking dimensional nesting. A.tolist() produces a list with 3 elements, each a 3 element list.

Rows, columns, planes are useful ways of talking about arrays, but are not a formal part of their definition.