Summary  
Explains L1 and L2 vector norms to measure vector magnitude and shows how to implement code to compute these norms.

General domain of usage  
Machine learning

**Video Overview:**  
This video introduces the concept of vector magnitude and explores two important vector norms: the L1 norm (Manhattan norm) and the L2 norm (Euclidean norm). You will learn how these norms are calculated, what they represent geometrically, and why they are significant in data science, including their roles in measuring distances and regularization techniques.

Understanding the size or "length" of a vector is essential in linear algebra and data science. The most common way to measure a vector's magnitude is the **Euclidean norm**, also called the **L2 norm**. For a vector $$v = [v₁, v₂, ..., vₙ]$$, the L2 norm calculates the straight-line distance from the origin to the point represented by the vector in n-dimensional space. The formula for the L2 norm is:

$$
||v||₂ = \sqrt{v₁² + v₂² + ... + vₙ²}
$$

Another important norm is the **L1 norm**, also known as the **Manhattan norm**. This norm sums the absolute values of the vector's components:

$$
||v||₁ = |v₁| + |v₂| + ... + |vₙ|
$$

These norms are widely used in data science. The **L2 norm** is often used for measuring Euclidean distance and in regularization (like Ridge regression), while the **L1 norm** is used in scenarios where sparsity is important (like Lasso regression).

import numpy as np

# Define a vector
v = np.array([3, -4, 1])

# Compute the L2 (Euclidean) norm
l2_norm = np.sqrt(np.sum(v ** 2))
print("L2 norm:", l2_norm)

# Compute the L1 norm
l1_norm = np.sum(np.abs(v))
print("L1 norm:", l1_norm)

In the code above, you define a vector `v` with three components. To find the **L2 norm**, you square each component, sum them, and then take the square root. This gives you the Euclidean distance from the origin to the point `[3, -4, 1]`. The **L1 norm** is calculated by summing the absolute values of each component, which measures the total "taxicab" or "Manhattan" distance.

The **L2 norm** is sensitive to large values in any component, as squaring amplifies their effect. This is suitable when you want to penalize large deviations strongly or measure straight-line distances. The **L1 norm** treats each component equally, making it robust to outliers and useful when you want to encourage sparsity in solutions, such as in feature selection for machine learning models. Choosing between these norms depends on your specific use case and the properties you want to emphasize in your data analysis or model.

Which statement best describes the difference between the L1 and L2 norms of a vector?

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Vector Norms and Magnitude