Creating vectors is as simple as using the function array  to convert a list to an array:

v = array([1.,2.,3.])

The object v  is now a vector that behaves much like a vector in linear algebra. We have already emphasized the differences with the list object in Python (refer to section Arrays in Chapter 3, Containers Type). Here are some illustrations of the basic linear algebra operations on vectors:

# two vectors with three components
v1 = array([1., 2., 3.])
v2 = array([2, 0, 1.])

# scalar multiplications/divisions
2*v1 # array([2., 4., 6.])
v1/2 # array([0.5, 1., 1.5])

# linear combinations
3*v1 # array([ 3., 6., 9.])
3*v1 + 2*v2 # array([ 7., 6., 11.])

# norm
from scipy.linalg import norm
norm(v1) # 3.7416573867739413
# scalar product
dot(v1, v2) # 5.
v1 @ v2 # 5 ; alternative formulation

Note that all basic arithmetic operations are performed elementwise:

# elementwise operations:
v1 * v2 # array([2., 0., 3.])
v2 / v1 # array([2.,0.,.333333])
v1 - v2 # array([-1., 2., 2.])
v1 + v2 # array([ 3., 2., 4.])

Some functions act elementwise on arrays as well:

cos(v1) # cosine, elementwise: array([ 0.5403,
                                 -0.4161, -0.9899])

This subject will be covered in the section Functions Acting on Arrays.

A matrix is created in a similar way to a vector, but from a list of lists instead:

M = array([[1.,2],[0.,1]])

To create a row matrix containing the same data as the vector v = array([1., 2., 1.]), we do this:

R = array([[1.,2.,1.]]) # notice the double brackets: 
                        # this is a matrix
shape(R)                # (1,3): this is a row matrix

The corresponding column matrix is obtained by the method reshape:

C = array([1., 2., 1.]).reshape(3, 1)
shape(C) # (3,1): this is a column matrix

Indexing and slicing are similar to that of a list. The main difference is that there may be several indexes or slices when the array is a matrix. The subject will be covered in depth in section Array indexing; here, we just give some illustrating examples of indexing and slicing:

v = array([1., 2., 3])
M = array([[1., 2],[3., 4]])

v[0] # works as for lists
v[1:] # array([2., 3.])

M[0, 0] # 1.
M[1:] # returns the matrix array([[3., 4]])
M[1] # returns the vector array([3., 4.])

# access
v[0] # 1.
v[0] = 10

# slices
v[:2] # array([10., 2.])
v[:2] = [0, 1] # now v == array([0., 1., 3.])
v[:2] = [1, 2, 3] # error!

The essential operator that performs most of the usual operations of linear algebra is the Python function dot. It is used for matrix-vector multiplications:

dot(M, v) # matrix vector multiplication; returns a vector
M @ v # alternative formulation

It may be used to compute a scalar product between two vectors:

dot(v, w) # scalar product; the result is a scalar
v @ w # alternative formulation

Lastly, it is used to compute matrix-matrix products:

dot(M, N) # results in a matrix
M @ N # alternative formulation

Solving a linear system

If A is a matrix and b is a vector, you can solve the linear equation:

Using the solve method, which has this syntax:

from scipy.linalg import solve
x = solve(A, b)

For example, we want to solve:

Here is the solution for the preceding equation:

from scipy.linalg import solve
A = array([[1., 2.], [3., 4.]])
b = array([1., 4.])
x = solve(A, b)
allclose(dot(A, x), b) # True
allclose(A @ x, b) # alternative formulation

The command allclose is used here to compare two vectors. If they are close enough to each other, this command returns True. Optionally a tolerance value can be set. For more methods related to linear equations systems, refer to section Linear algebra methods in SciPy.