Solving linear systems (solve) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of solving a 2x2 linear system

What is the output of this code that solves a simple 2x2 linear system?

SciPy

import numpy as np
from scipy.linalg import solve
A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])
x = solve(A, b)
print(x)

A [2. 3.]

B [3. 2.]

C [1. 4.]

D [4. 1.]

💡 Hint

Remember to multiply matrix A by vector x to get vector b.

The system is 3x + y = 9 and x + 2y = 8. Solving gives x=2, y=3.

❓ data_output

intermediate

Number of solutions for a singular matrix

What happens when you try to solve a linear system with a singular matrix using scipy.linalg.solve?

SciPy

import numpy as np
from scipy.linalg import solve
A = np.array([[1, 2], [2, 4]])
b = np.array([5, 10])
x = solve(A, b)
print(x)

A [1. 2.]

B [5. 0.]

C [0. 5.]

D LinAlgError: Singular matrix

💡 Hint

Check if the matrix can be inverted.

Matrix A is singular (second row is multiple of first), so solve raises LinAlgError.

🔧 Debug

advanced

Identify the error in solving a system with mismatched dimensions

What error does this code produce and why?

SciPy

import numpy as np
from scipy.linalg import solve
A = np.array([[1, 2], [3, 4]])
b = np.array([5, 6, 7])
x = solve(A, b)
print(x)

A ValueError: incompatible dimensions

B LinAlgError: Singular matrix

C TypeError: unsupported operand type(s)

D IndexError: index out of bounds

💡 Hint

Check the shape of A and b before solving.

Matrix A is 2x2 but vector b has length 3, so dimensions don't match for solving.

🧠 Conceptual

advanced

Effect of using scipy.linalg.solve vs numpy.linalg.solve

Which statement correctly describes the difference between scipy.linalg.solve and numpy.linalg.solve?

A numpy.linalg.solve is faster and more accurate than scipy.linalg.solve

B scipy.linalg.solve can handle more matrix types and offers more options than numpy.linalg.solve

C Both functions are identical and interchangeable without any difference

D scipy.linalg.solve only works for sparse matrices while numpy.linalg.solve works for dense matrices

💡 Hint

Think about the libraries' scope and features.

scipy.linalg.solve is built on top of LAPACK and supports more options and matrix types than numpy.linalg.solve.

🚀 Application

expert

Solving a system with a large sparse matrix efficiently

You have a large sparse matrix A and vector b. Which approach is best to solve Ax = b efficiently?

A Use numpy.linalg.solve directly on sparse matrix

B Convert A to dense and use scipy.linalg.solve

C Use scipy.sparse.linalg.spsolve for sparse matrix solving

D Use a for loop to solve each equation separately

💡 Hint

Sparse matrices need special solvers to save memory and time.

scipy.sparse.linalg.spsolve is designed for sparse matrices and is efficient for large systems.