outer() for outer operations in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of numpy.outer with simple vectors

What is the output of the following code using numpy.outer?

NumPy

import numpy as np

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

A

[[3 4]
 [6 8]]

B

[[3 6]
 [4 8]]

C

[[4 3]
 [8 6]]

D

[[1 2]
 [3 4]]

💡 Hint

Remember that np.outer multiplies each element of the first array by each element of the second array.

The np.outer function computes the outer product of two vectors. For vectors x and y, the result is a matrix where each element is x[i] * y[j]. Here, 1*3=3, 1*4=4, 2*3=6, and 2*4=8.

❓ data_output

intermediate

Shape of the output from np.outer with 1D arrays

Given two 1D numpy arrays a with shape (4,) and b with shape (3,), what is the shape of np.outer(a, b)?

NumPy

import numpy as np

a = np.arange(4)
b = np.arange(3)
result = np.outer(a, b)
print(result.shape)

A (3, 3)

B (4, 3)

C (4, 4)

D (3, 4)

💡 Hint

The outer product shape is (len(a), len(b)).

The outer product of two 1D arrays with lengths m and n results in a 2D array with shape (m, n). Here, a has length 4 and b has length 3, so the shape is (4, 3).

❓ visualization

advanced

Visualizing the outer product matrix

Which option shows the correct heatmap visualization of np.outer([1, 2, 3], [4, 5])?

NumPy

import numpy as np
import matplotlib.pyplot as plt

matrix = np.outer([1, 2, 3], [4, 5])
plt.imshow(matrix, cmap='viridis')
plt.colorbar()
plt.show()

A A 3x2 heatmap with values [[4, 5], [8, 10], [12, 15]] increasing top to bottom and left to right

B A 2x3 heatmap with values [[4, 8, 12], [5, 10, 15]] increasing left to right and top to bottom

C A 3x2 heatmap with values [[1, 2], [3, 4], [5, 6]] increasing top to bottom and left to right

D A 2x3 heatmap with values [[1, 3, 5], [2, 4, 6]] increasing left to right and top to bottom

💡 Hint

Check the shape and values of the outer product matrix before visualizing.

The outer product of vectors [1, 2, 3] and [4, 5] is a 3x2 matrix where each element is the product of elements from the first and second vectors. The values are [[4, 5], [8, 10], [12, 15]]. The heatmap will show these increasing values from top to bottom and left to right.

🧠 Conceptual

advanced

Understanding the difference between np.outer and np.dot

Which statement correctly describes the difference between np.outer and np.dot when applied to 1D numpy arrays?

A <code>np.outer</code> and <code>np.dot</code> produce the same output for 1D arrays.

B <code>np.outer</code> performs element-wise multiplication, while <code>np.dot</code> performs matrix multiplication.

C <code>np.outer</code> produces a scalar, while <code>np.dot</code> produces a matrix of pairwise products.

D <code>np.outer</code> produces a matrix of all pairwise products, while <code>np.dot</code> produces a single scalar sum of element-wise products.

💡 Hint

Think about the shapes of the outputs and what each function computes.

np.outer computes the outer product, resulting in a matrix where each element is the product of elements from the two vectors. np.dot computes the dot product, which for 1D arrays is the sum of element-wise products, resulting in a scalar.

🔧 Debug

expert

Identify the error in using np.outer with 2D arrays

What error will the following code produce and why?

import numpy as np

x = np.array([[1, 2], [3, 4]])
y = np.array([[5, 6], [7, 8]])
result = np.outer(x, y)
print(result.shape)

NumPy

import numpy as np

x = np.array([[1, 2], [3, 4]])
y = np.array([[5, 6], [7, 8]])
result = np.outer(x, y)
print(result.shape)

A ValueError due to shape mismatch between 2D arrays

B TypeError because np.outer only accepts 1D arrays

C No error; output shape is (4, 4) because inputs are flattened before outer product

D AttributeError because np.outer is not defined for numpy arrays

💡 Hint

Check how np.outer handles multi-dimensional arrays internally.

np.outer flattens any input arrays before computing the outer product. So, 2D arrays are treated as 1D arrays of their elements. Here, both x and y are flattened to length 4, so the output shape is (4, 4).