Broadcasting for outer products in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of broadcasting outer product with 1D arrays

What is the output of this code snippet using NumPy broadcasting to compute an outer product?

NumPy

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5])
result = x[:, None] * y
print(result)

A

[[ 4  5]
 [ 8 10]
 [12 15]]

B

[[ 4  8 12]
 [ 5 10 15]]

C

[[ 4  5  6]
 [ 8 10 12]
 [12 15 18]]

D

[[ 5  4]
 [10  8]
 [15 12]]

💡 Hint

Remember that x[:, None] reshapes x to a column vector to enable broadcasting with y.

The expression x[:, None] reshapes x from shape (3,) to (3,1). Multiplying by y with shape (2,) broadcasts y to (1,2), resulting in a (3,2) array where each element is the product of elements from x and y.

❓ data_output

intermediate

Shape of result from broadcasting outer product

Given two arrays x with shape (4,) and y with shape (3,), what is the shape of the result when computing x[:, None] * y using NumPy broadcasting?

NumPy

import numpy as np
x = np.arange(4)
y = np.arange(3)
result = x[:, None] * y
print(result.shape)

A (4, 3)

B (3, 4)

C (4,)

D (3,)

💡 Hint

Think about how reshaping x to (4,1) and y to (3,) broadcasts to a 2D array.

x[:, None] reshapes x to (4,1). y has shape (3,). Broadcasting y to (1,3) and multiplying results in shape (4,3).

🔧 Debug

advanced

Identify the error in broadcasting outer product code

What error does this code raise when trying to compute an outer product using broadcasting?

NumPy

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

A IndexError: index out of bounds

B TypeError: unsupported operand type(s) for *: 'int' and 'list'

C No error, output is [4 10 15]

D ValueError: operands could not be broadcast together with shapes (3,) (2,)

💡 Hint

Check the shapes of x and y before multiplication.

x has shape (3,) and y has shape (2,). They cannot be broadcast together directly for element-wise multiplication, causing a ValueError.

🚀 Application

advanced

Use broadcasting to compute outer product without loops

Which code snippet correctly computes the outer product of two 1D NumPy arrays x and y without using loops?

A result = x * y[:, None]

B result = x[:, None] * y

C result = np.dot(x, y)

D result = np.outer(x, y) + 1

💡 Hint

Remember the shape changes needed for broadcasting to work as outer product.

x[:, None] reshapes x to a column vector, y stays as a row vector, so multiplication broadcasts to outer product. np.dot computes inner product, np.outer adds 1 changes values.

🧠 Conceptual

expert

Understanding broadcasting rules for outer products

Which statement best explains why x[:, None] * y computes the outer product of 1D arrays x and y in NumPy?

A x[:, None] transposes x, y is broadcast to match x's shape, resulting in a dot product instead of outer product.

B x[:, None] flattens x to 1D, y is reshaped to 2D, so broadcasting sums the arrays to produce the outer product.

C x[:, None] reshapes x to a 2D column vector, y remains 1D, so broadcasting expands y to a row vector, enabling element-wise multiplication to produce a 2D outer product.

D x[:, None] duplicates x elements, y duplicates y elements, and their multiplication produces a scalar product.

💡 Hint

Focus on how reshaping changes array dimensions and how broadcasting works with those shapes.

Reshaping x to (n,1) and y to (m,) allows broadcasting to (n,m) for element-wise multiplication, which is the definition of outer product.