Broadcasting for outer products in NumPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to create an outer product of two 1D arrays using broadcasting.

NumPy

import numpy as np

x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
outer = x[:, [1]] * y
print(outer)

Drag options to blanks, or click blank then click option'

Areshape(3, 1)

Bnp.newaxis

CNone

Dreshape(1, 3)

Attempts:

3 left

💡 Hint

Common Mistakes

Using reshape(1, 3) instead of adding a new axis.

Trying to multiply without changing dimensions, causing shape errors.

Using np.newaxis adds a new axis to x, making it a column vector. This allows broadcasting to multiply it with y as a row vector, producing the outer product.

2fill in blank

medium

Complete the code to compute the outer product of two arrays using broadcasting with shape manipulation.

NumPy

import numpy as np

a = np.array([2, 3])
b = np.array([5, 7, 11])
result = a.[1] * b
print(result)

Drag options to blanks, or click blank then click option'

Areshape(2, 1)

Breshape(1, 2)

Creshape(3, 1)

Dreshape(1, 3)

Attempts:

3 left

💡 Hint

Common Mistakes

Reshaping a incorrectly as (1, 2) which makes it a row vector.

Not reshaping a at all, causing shape mismatch.

Reshaping a to (2, 1) makes it a column vector. Multiplying by b (shape (3,)) broadcasts b as a row vector, producing a (2, 3) outer product.

3fill in blank

hard

Fix the error in the code to correctly compute the outer product using broadcasting.

NumPy

import numpy as np

x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
outer = x.[1] * y
print(outer)

Drag options to blanks, or click blank then click option'

Areshape(3, 1)

Breshape(3, 3)

Creshape(1, 3)

Dflatten()

Attempts:

3 left

💡 Hint

Common Mistakes

Using reshape(3, 3) which changes the data shape incorrectly.

Using flatten() which does not add a new axis.

Reshaping x to (3, 1) makes it a column vector. This allows broadcasting with y (shape (3,)) to compute the outer product.

4fill in blank

hard

Fill both blanks to create an outer product using broadcasting with reshaping.

NumPy

import numpy as np

p = np.array([7, 8])
q = np.array([1, 2, 3])
outer_product = p.[1] * q.[2]
print(outer_product)

Drag options to blanks, or click blank then click option'

Areshape(2, 1)

Breshape(1, 3)

Cflatten()

Dreshape(3, 1)

Attempts:

3 left

💡 Hint

Common Mistakes

Using flatten() which removes dimensions instead of reshaping.

Swapping the shapes causing broadcasting errors.

Reshape p to (2, 1) to make it a column vector, and q to (1, 3) to make it a row vector. Multiplying them broadcasts to a (2, 3) outer product.

5fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each word to its length only if the length is greater than 3.

NumPy

words = ['data', 'science', 'ai', 'ml']
lengths = {word: {BLANK_2}} for word in words if {{BLANK_2}}

Drag options to blanks, or click blank then click option'

Blen(word)

Clen(word) > 3

Dword

Attempts:

3 left

💡 Hint

Common Mistakes

Adding a colon after the word key in the dictionary comprehension.

Using the word itself as the value instead of its length.

Not filtering words by length.

The dictionary comprehension uses word as the key (no extra syntax needed, so blank 1 is empty), len(word) as the value, and filters words with length greater than 3.

Practice

1. What does broadcasting allow you to do in NumPy when working with arrays of different shapes?

easy

A. Convert arrays into lists for easier manipulation

B. Perform element-wise operations without explicit loops by automatically expanding shapes

C. Sort arrays in ascending order automatically

D. Create new arrays filled with zeros matching the shape of another array

Solution

1. Step 1: Understand broadcasting concept

   Broadcasting lets NumPy perform operations on arrays of different shapes by expanding the smaller array's shape without copying data.

2. Step 2: Identify what broadcasting enables

   This means you can do element-wise operations without writing loops explicitly.

3. Final Answer:

   Perform element-wise operations without explicit loops by automatically expanding shapes -> Option B

4. Quick Check:

   Broadcasting = element-wise operations without loops [OK]

Hint: Broadcasting expands shapes to match for element-wise ops [OK]

Common Mistakes:

• Thinking broadcasting converts arrays to lists
• Confusing broadcasting with sorting
• Assuming broadcasting creates new zero arrays

2. Which of the following is the correct way to reshape a 1D NumPy array a to a column vector for broadcasting?

easy

A. a[np.newaxis, :]

B. a.reshape(1, -1)

C. a[:, np.newaxis]

D. a.flatten()

Solution

1. Step 1: Understand reshaping to column vector

   A column vector has shape (n, 1). Using np.newaxis adds a new axis. a[:, np.newaxis] converts shape (n,) to (n,1).

2. Step 2: Check other options

   a.reshape(1, -1) makes a row vector (1, n). a[np.newaxis, :] also makes a row vector. a.flatten() flattens array to 1D, no new axis.

3. Final Answer:

   a[:, np.newaxis] -> Option C

4. Quick Check:

   Column vector = a[:, np.newaxis] [OK]

Hint: Use a[:, np.newaxis] to add a column axis [OK]

Common Mistakes:

• Using reshape(1, -1) creates a row, not column
• Confusing row and column vector shapes
• Using flatten removes dimensions instead of adding

3. What is the output of the following code?

import numpy as np
x = np.array([1, 2])
y = np.array([3, 4, 5])
result = x[:, np.newaxis] * y
print(result)

medium

A. [[3 6]
[4 8]
[5 10]]

B. [[3 4]
[5 6]]

C. [3 4 5 6 8 10]

D. [[3 4 5]
[6 8 10]]

Solution

1. Step 1: Reshape x for broadcasting

   x[:, np.newaxis] changes x from shape (2,) to (2,1): [[1], [2]].

2. Step 2: Multiply with y

   y has shape (3,), broadcasted to (1,3): [3,4,5]. Multiplying (2,1) * (1,3) results in (2,3) array:
   [[1*3, 1*4, 1*5],
   [2*3, 2*4, 2*5]] = [[3,4,5],[6,8,10]]

3. Final Answer:

   [[3 4 5] [6 8 10]] -> Option D

4. Quick Check:

   Outer product shape (2,3) = [[3 4 5], [6 8 10]] [OK]

Hint: Reshape first array to (n,1) to multiply outer product [OK]

Common Mistakes:

• Not reshaping x, causing shape mismatch
• Confusing row and column broadcasting
• Expecting 1D output instead of 2D

4. The following code is intended to compute the outer product of two arrays a and b. What is the error?

import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5])
result = a * b
print(result)

medium

A. Shape mismatch error because arrays have incompatible shapes for element-wise multiplication

B. Syntax error due to missing parentheses

C. Result is a scalar instead of an array

D. No error; code runs correctly

Solution

1. Step 1: Check shapes of a and b

   a has shape (3,), b has shape (2,). They cannot be multiplied element-wise directly because shapes differ and are not broadcast-compatible.

2. Step 2: Identify error cause

   Multiplying arrays of incompatible shapes without reshaping causes a ValueError for shape mismatch.

3. Final Answer:

   Shape mismatch error because arrays have incompatible shapes for element-wise multiplication -> Option A

4. Quick Check:

   Shape mismatch error = incompatible shapes [OK]

Hint: Check shapes before multiplying; reshape if needed [OK]

Common Mistakes:

• Assuming arrays multiply element-wise without matching shapes
• Thinking code has syntax errors
• Expecting scalar output from array multiplication

5. You want to compute the outer product of two arrays x = np.array([1, 2, 3]) and y = np.array([4, 5]) using broadcasting without explicit loops. Which code snippet correctly does this and produces a (3, 2) shaped result?

hard

A. x[:, np.newaxis] * y

B. np.dot(x, y)

C. x * y[:, np.newaxis]

D. np.multiply.outer(x, y)

Solution

1. Step 1: Understand shapes and broadcasting

   x has shape (3,), y has shape (2,). To get outer product shape (3,2), reshape x to (3,1) and multiply by y (2,) broadcasted to (1,2).

2. Step 2: Evaluate options

   np.multiply.outer(x, y) does exist as a function but returns a (3,2) array; however, the question asks for broadcasting without explicit loops and np.multiply.outer is a function call, not broadcasting syntax. np.dot(x, y) computes dot product, resulting in scalar. x * y[:, np.newaxis] reshapes y to (2,1), resulting in (3,) * (2,1) which broadcasts to (2,3) (swapped dimensions), not (3,2). x[:, np.newaxis] * y reshapes x to (3,1) and multiplies by y, producing (3,2).

3. Final Answer:

   x[:, np.newaxis] * y -> Option A

4. Quick Check:

   Broadcast x to (3,1) and multiply y (2,) = (3,2) [OK]

Hint: Reshape first array to (n,1) to get outer product shape [OK]

Common Mistakes:

• Using np.dot instead of element-wise multiplication
• Confusing np.multiply.outer with broadcasting syntax
• Swapping axes by reshaping wrong array