Summary  
This chapter covers representing vectors as lists in Python and implementing element-wise vector addition, subtraction, and scalar multiplication using list comprehensions.

General domain of usage  
Computer graphics

This video introduces the concept of **vectors**, their mathematical notation, and their importance in real-world applications. You will see visual examples such as arrows in space, velocities, and forces to illustrate vectors’ **direction** and **magnitude**. The video also shows how vectors are represented and manipulated in `Python`, connecting mathematical ideas to practical programming.

Understanding vectors is the first step to mastering linear algebra. A **vector** is a mathematical object that has both magnitude (size) and direction. Vectors are often written as an ordered list of numbers, like $$[2, 3]$$ or $$[1, -4, 5]$$. The number of elements in a vector is called its **dimension**, for example, $$[2, 3]$$ is a 2-dimensional vector, while $$[1, -4, 5]$$ is 3-dimensional.

Vectors appear everywhere in the real world. You can think of a vector as representing a force pushing an object, a velocity describing movement through space, or a position on a map. In computer graphics, vectors describe points and directions; in physics, they describe forces and velocities. In data science, they represent data points with many features.

# Creating vectors in Python using lists
vector_a = [2, 3]
vector_b = [4, 1]

# Vector addition
vector_sum = [a + b for a, b in zip(vector_a, vector_b)]

# Vector subtraction
vector_diff = [a - b for a, b in zip(vector_a, vector_b)]

# Scalar multiplication (multiply each element by a number)
scalar = 3
vector_scaled = [scalar * a for a in vector_a]

print("Vector A:", vector_a)
print("Vector B:", vector_b)
print("A + B:", vector_sum)
print("A - B:", vector_diff)
print("3 * A:", vector_scaled)

In this code, you create two vectors, `vector_a` and `vector_b`, as Python lists. Each list contains two numbers, making them 2-dimensional vectors. To add or subtract vectors, you combine their corresponding elements. This is done using a list comprehension with the `zip` function, which pairs up the elements from each vector.

For vector addition, `[a + b for a, b in zip(vector_a, vector_b)]` adds the first elements of `vector_a` and `vector_b` to get the first element of the result, and does the same for the second elements. Vector subtraction works the same way, but with subtraction.

Scalar multiplication means multiplying every element of a vector by the same number. In the code, `[scalar * a for a in vector_a]` multiplies each element of `vector_a` by `3`.

Python lists allow you to represent vectors and perform these operations, but you must handle the arithmetic element by element. This approach helps you see the structure of vector operations, which is essential for understanding more advanced linear algebra topics.

Which of the following statements about vectors and their operations is **correct** based on the code and explanations above?

Establish core mathematical intuition for data science by manipulating vectors, multiplying matrices, solving linear systems, and computing eigenvalues for dimensionality reduction, all utilizing Python.

All core linear algebra concepts for data science in one section, with each chapter building on the previous to develop a strong mathematical foundation using Python.

Vectors and Their Operations