np.linalg.norm() for vector norms in NumPy

Introduction

We use np.linalg.norm() to find the length or size of a vector. It helps us understand how big or small a vector is in simple terms.

To measure the distance of a point from the origin in space.

To find how far apart two points are by subtracting their coordinates and measuring the vector length.

To check the size of a vector in machine learning, like weights or feature vectors.

To normalize vectors so they have length 1, making comparisons easier.

Syntax

NumPy

np.linalg.norm(x, ord=None, axis=None, keepdims=False)

x is the input vector or array.

ord specifies the type of norm (default is Euclidean norm).

Examples

Calculate the Euclidean length of vector [3, 4], which is 5.

NumPy

import numpy as np
v = np.array([3, 4])
norm = np.linalg.norm(v)

Calculate the 1-norm (sum of absolute values) of the vector.

NumPy

np.linalg.norm([1, -1, 1], ord=1)

Calculate the infinity norm (maximum absolute value) of the vector.

NumPy

np.linalg.norm([2, -3, 6], ord=np.inf)

Sample Program

This program shows how to calculate different norms of the same vector. It prints the vector and its Euclidean, 1-norm, and infinity norm values.

NumPy

import numpy as np

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

# Calculate Euclidean norm (default)
euclidean_norm = np.linalg.norm(vector)

# Calculate 1-norm (sum of absolute values)
one_norm = np.linalg.norm(vector, ord=1)

# Calculate infinity norm (max absolute value)
inf_norm = np.linalg.norm(vector, ord=np.inf)

print(f"Vector: {vector}")
print(f"Euclidean norm (length): {euclidean_norm}")
print(f"1-norm (sum of abs): {one_norm}")
print(f"Infinity norm (max abs): {inf_norm}")

Important Notes

The default norm is the Euclidean norm, which is like the straight-line distance.

You can use different ord values to get other types of norms.

Works for vectors and matrices, but here we focus on vectors.

Summary

np.linalg.norm() finds the size or length of a vector.

Default is Euclidean norm, but you can choose others like 1-norm or infinity norm.

Useful for measuring distances and normalizing vectors in data science.

Practice

1. What does np.linalg.norm() calculate by default when given a vector?

easy

A. The maximum element in the vector

B. The sum of all vector elements

C. The Euclidean length (distance) of the vector

D. The product of all vector elements

Solution

1. Step 1: Understand the default behavior of np.linalg.norm()

   By default, np.linalg.norm() calculates the Euclidean norm, which is the straight-line distance from the origin to the point represented by the vector.

2. Step 2: Compare with other options

   The sum, max, and product are different operations and not what np.linalg.norm() returns by default.

3. Final Answer:

   The Euclidean length (distance) of the vector -> Option C

4. Quick Check:

   Default norm = Euclidean length [OK]

Hint: Default norm is Euclidean distance, not sum or max [OK]

Common Mistakes:

• Confusing norm with sum of elements
• Thinking norm returns max element
• Assuming norm multiplies elements

2. Which of the following is the correct syntax to compute the 1-norm (sum of absolute values) of a vector v using np.linalg.norm()?

easy

A. np.linalg.norm(v, order=1)

B. np.linalg.norm(v, ord=1)

C. np.linalg.norm(v, norm=1)

D. np.linalg.norm(v, p=1)

Solution

1. Step 1: Recall the parameter name for norm order

   The parameter to specify the norm order in np.linalg.norm() is ord, not order, norm, or p.

2. Step 2: Check the correct syntax

   Using ord=1 correctly computes the 1-norm, which sums the absolute values of vector elements.

3. Final Answer:

   np.linalg.norm(v, ord=1) -> Option B

4. Quick Check:

   Use ord=1 for 1-norm [OK]

Hint: Use ord=1 to specify 1-norm in np.linalg.norm() [OK]

Common Mistakes:

• Using 'order' instead of 'ord'
• Using 'norm' or 'p' as parameter names
• Omitting the ord parameter for 1-norm

3. What is the output of the following code?

import numpy as np
v = np.array([3, 4])
norm_val = np.linalg.norm(v)
print(norm_val)

medium

A. 5.0

B. 7

C. 12

D. 1

Solution

1. Step 1: Calculate the Euclidean norm of vector [3, 4]

   The Euclidean norm is sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.0.

2. Step 2: Confirm the printed output

   The code prints the norm value, which is 5.0.

3. Final Answer:

   5.0 -> Option A

4. Quick Check:

   Euclidean norm of [3,4] = 5.0 [OK]

Hint: Euclidean norm of (3,4) is 5 by Pythagoras [OK]

Common Mistakes:

• Adding elements instead of squaring and summing
• Forgetting to take square root
• Confusing norm with sum or max

4. The following code throws an error. What is the mistake?

import numpy as np
v = np.array([1, -2, 3])
norm_val = np.linalg.norm(v, order=2)
print(norm_val)

medium

A. The parameter name should be 'ord' not 'order'

B. The vector contains negative values which cause error

C. np.linalg.norm() does not accept a second argument

D. The vector must be a list, not a numpy array

Solution

1. Step 1: Identify the parameter name error

   The parameter to specify norm order is ord, not order. Using order causes a TypeError.

2. Step 2: Confirm other options are incorrect

   Negative values are allowed, np.linalg.norm accepts second argument ord, and numpy arrays are valid inputs.

3. Final Answer:

   The parameter name should be 'ord' not 'order' -> Option A

4. Quick Check:

   Use ord=2, not order=2 [OK]

Hint: Use ord= for norm order, not order= [OK]

Common Mistakes:

• Using 'order' instead of 'ord'
• Thinking negative values cause error
• Believing np.linalg.norm takes only one argument

5. You have a dataset of 2D points stored as rows in a numpy array points. You want to normalize each point to have length 1 (unit vector). Which code correctly does this using np.linalg.norm()?

hard

A. normalized = points / np.linalg.norm(points, ord=1, axis=1)

B. normalized = points / np.linalg.norm(points, axis=0)

C. normalized = points / np.linalg.norm(points)

D. normalized = points / np.linalg.norm(points, axis=1, keepdims=True)

Solution

1. Step 1: Calculate norms along rows with correct shape

   Using np.linalg.norm(points, axis=1, keepdims=True) computes the Euclidean norm for each row and keeps the result as a column vector, allowing correct broadcasting for division.

2. Step 2: Normalize each point by dividing by its norm

   Dividing points by the norms with matching shape normalizes each row vector to length 1.

3. Final Answer:

   normalized = points / np.linalg.norm(points, axis=1, keepdims=True) -> Option D

4. Quick Check:

   Use keepdims=True for correct broadcasting [OK]

Hint: Use keepdims=True to keep norm shape for division [OK]

Common Mistakes:

• Using axis=0 instead of axis=1 for row-wise normalization
• Using norm of whole array instead of per row
• Using ord=1 instead of default Euclidean norm