Now that we know how to use NumPy arrays to create vectors and matrices, we can learn how to perform various linear algebra operations using Python.

To multiply a vector or matrix by a scalar, we use inbuilt Python multiplication between the NumPy array and the scalar:

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

Written out mathematically, this is:

$4 \begin{bmatrix}
1 & 2 \\
3 & 4 
\end{bmatrix}=
\begin{bmatrix}
4 & 8 \\
12 & 16 
\end{bmatrix}$

Therefore, our output in Python is:

[[ 4  8]
 [12 16]]

To add equally sized vectors or matrices, we can again use inbuilt Python addition between the NumPy arrays.

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

Written out mathematically, this is:

$\begin{bmatrix}
1 & 2 \\
3 & 4 
\end{bmatrix}+
\begin{bmatrix}
-4 & -3 \\
-2 & -1 
\end{bmatrix}=
\begin{bmatrix}
-3 & -1 \\
1 & 3 
\end{bmatrix}$

Therefore, our output in Python is:

[[ -3 -1]
 [1 3]]

Vector dot products can be computed using the np.dot() function:

v = np.array([-1,-2,-3])
u = np.array([2,2,2])
np.dot(v,u)

Written out mathematically, this is:

$\begin{bmatrix}
-1 \\
-2\\ 
-3 \\
\end{bmatrix}\cdot
\begin{bmatrix}
2 \\
2 \\
2
\end{bmatrix} =
-12$

Therefore, our output in Python is:

-12

Matrix multiplication is computed using either the np.matmul() function or using the @ symbol as shorthand. It is important to note that using the typical Python multiplication symbol * will result in an elementwise multiplication instead.

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

# one way to matrix multiply
np.matmul(A,B)
# another way to matrix multiply
[email protected]

Written out mathematically, this is:

$\begin{bmatrix}
1 & 2 \\
3 & 4 
\end{bmatrix}\cdot
\begin{bmatrix}
-4 & -3 \\
-2 & -1 
\end{bmatrix}=
\begin{bmatrix}
-8 & -5 \\
-20 & -13 
\end{bmatrix}$

Therefore, our Python output is:

[[ -8  -5]
 [-20 -13]]