np.linalg.norm() for vector norms in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does np.linalg.norm() compute when applied to a vector?

np.linalg.norm() calculates the length or size of a vector, also called the vector's norm.

beginner

How do you calculate the Euclidean norm (length) of a vector v using np.linalg.norm()?

Use np.linalg.norm(v). This gives the straight-line distance from the origin to the point represented by v.

intermediate

What parameter do you use in np.linalg.norm() to calculate the Manhattan norm (sum of absolute values)?

Set ord=1 in np.linalg.norm() to get the Manhattan norm, which sums the absolute values of vector elements.

intermediate

What does np.linalg.norm(v, ord=np.inf) compute?

It computes the maximum absolute value among the elements of vector v, called the infinity norm.

advanced

If you want to calculate the squared Euclidean norm of a vector v efficiently, what is a good approach?

Calculate np.dot(v, v) instead of np.linalg.norm(v)**2 to avoid the square root and save computation.

What is the default norm calculated by np.linalg.norm() when applied to a vector?

A Euclidean norm (L2 norm)

B Manhattan norm (L1 norm)

C Infinity norm

D Squared Euclidean norm

The default is the Euclidean norm, which is the square root of the sum of squares of the vector elements.

Which ord value in np.linalg.norm() gives the Manhattan norm?

A np.inf

B 0

C 2

D 1

Setting ord=1 calculates the Manhattan norm, the sum of absolute values.

What does np.linalg.norm(v, ord=np.inf) return?

A Sum of vector elements

B Maximum absolute value

C Minimum absolute value

D Euclidean norm

It returns the maximum absolute value among the vector elements, known as the infinity norm.

How can you compute the squared Euclidean norm of a vector v efficiently?

A np.linalg.norm(v)**2

B np.sum(v)

C np.dot(v, v)

D np.linalg.norm(v, ord=2)

Using np.dot(v, v) avoids the square root and is faster for squared norm.

What will np.linalg.norm([3, 4]) return?

A 5

B 25

C 12

D 7

The Euclidean norm is sqrt(3² + 4²) = 5.

Explain how to use np.linalg.norm() to calculate different types of vector norms and give examples.

Think about the 'ord' parameter and what each norm means.

Describe why and how you might compute the squared Euclidean norm of a vector without using np.linalg.norm().

Consider the math behind Euclidean norm and dot product.

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