The core of using NumPy effectively for linear algebra is using NumPy arrays. NumPy arrays are n-dimensional array data structures that can be used to represent both vectors (1-dimensional array) and matrices (2-dimensional arrays).

A NumPy array is initialized using the np.array() function, and including a Python list argument or Python nested list argument for arrays with more than one dimension.

For example, the following creates a NumPy array representation of a vector:

v = np.array([1, 2, 3, 4, 5, 6])

We can also create a matrix, which is the equivalent of a two-dimensional NumPy array, using a nested Python list:

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

If we print this NumPy array, it will look like the following:

[[1 2]
 [3 4]]

Matrices can also be created by combining existing vectors using the np.column_stack() function:

v = np.array([-2,-2,-2,-2])
u = np.array([0,0,0,0])
w = np.array([3,3,3,3])

A = np.column_stack((v, u, w))
print(A)

Output:

[[-2  0  3]
 [-2  0  3]
 [-2  0  3]
 [-2  0  3]]

To access the shape of a matrix or vector once it’s been created as a NumPy array, we call the .shape attribute of the array variable:

A = np.array([[1,2],[3,4]])
print(A.shape)

Here, the output will be:

(2, 2)

since there are two rows and two columns.

To access individual elements in a NumPy array, we can index the array using square brackets. Unlike regular Python lists, we can index into all dimensions in a single square bracket, separating the dimension indices with commas.

Thus in order to index the element equal to 2 in matrix A, we can do the following:

A = np.array([[1,2],[3,4]])
print(A[0,1])

The first index accesses the specific row, while the second index accesses the specific column. Note that rows and columns are zero-indexed. In this example, the output will be the following:

2

We can also select a subset or entire dimension of a NumPy array using a colon. For example, if we want the entire second column of a matrix, we can index the second column and use an empty colon to select every row as such:

A = np.array([[1,2],[3,4]])
print(A[:,1])

Output:

[2 4]

Note: [2 4] is a column vector, but it outputs in the terminal horizontally.