In machine learning, vectors are essential for representing diverse types of data, including numerical, categorical, and text data.

They form the framework for fundamental operations like adding and multiplying with a scalar.

What is a Vector?

A vector is a tuple of one or more values, known as its components. Each component can be a number, category, or more abstract entities. In machine learning, vectors are commonly represented as one-dimensional arrays.

Types of Vectors

• Row Vector: Will have only one row.
• Column Vector: Comprising of only one column.

Play and experiment with the code to know about vectors. Here is the Python code:

# Define a row vector with 3 components
row_vector = [1, 2, 3]

# Define a column vector with 3 components
column_vector = [[1],
                 [2],
                 [3]]

# Print the vectors
print("Row Vector:", row_vector)
print("Column Vector:", column_vector)

Common Vector Operations in Machine Learning

Addition

Each corresponding element is added.

$$\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix}$$

Dot Product

Sum of the products of corresponding elements.

$$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} = 1 \times 4 + 2 \times 5 + 3 \times 6 = 32$$

Multiplying with a Scalar

Each element is multiplied by the scalar.

$$2 \times \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}$$

Length (Magnitude)

Euclidean distance is calculated by finding the square root of the sum of squares of individual elements.

$$\| \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}$$