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.

Practice

1. What does the scipy.linalg.solve function do in solving linear systems?

easy

A. It finds the exact solution vector for a system of linear equations.

B. It approximates the solution using random guesses.

C. It only checks if the system has any solution without solving.

D. It multiplies two matrices together.

Solution

1. Step 1: Understand the purpose of scipy.linalg.solve

   This function is designed to solve linear systems of the form Ax = b, where A is a matrix and b is a vector.

2. Step 2: Identify the output of the function

   The function returns the vector x that exactly satisfies the equation Ax = b.

3. Final Answer:

   It finds the exact solution vector for a system of linear equations. -> Option A

4. Quick Check:

   Function purpose = exact solution vector [OK]

Hint: Remember: solve() returns exact solutions, not guesses. [OK]

Common Mistakes:

• Thinking solve() approximates solutions
• Confusing solve() with matrix multiplication
• Assuming solve() only checks for solution existence

2. Which of the following is the correct way to import the solve function from scipy.linalg?

easy

A. from scipy.linalg import solve

B. import solve from scipy.linalg

C. from scipy import solve.linalg

D. import scipy.solve.linalg

Solution

1. Step 1: Recall Python import syntax for functions

   To import a specific function from a module, use from module import function.

2. Step 2: Match the correct module and function

   The function solve is inside the scipy.linalg module, so the correct import is from scipy.linalg import solve.

3. Final Answer:

   from scipy.linalg import solve -> Option A

4. Quick Check:

   Correct import syntax = from scipy.linalg import solve [OK]

Hint: Use 'from module import function' for specific imports. [OK]

Common Mistakes:

• Using 'import solve from ...' which is invalid syntax
• Trying to import submodules as functions
• Using dot notation incorrectly in import statements

3. What is the output of this code?

import numpy as np
from scipy.linalg import solve
A = np.array([[2, 1], [1, 3]])
b = np.array([10, 15])
x = solve(A, b)
print(x)

medium

A. [4. 3.]

B. [3. 4.]

C. [2. 5.]

D. [5. 2.]

Solution

1. Step 1: Set up the system of equations

   The matrix A and vector b represent the system:
   2x + y = 10
   x + 3y = 15

2. Step 2: Solve the system manually

   Solving manually:
   Multiply first equation by 3: 6x + 3y = 30
   Subtract second equation: 5x = 15 -> x = 3
   y = 10 - 2*3 = 4
   Thus, x = [3. 4.], which matches the output of solve(A, b).

3. Final Answer:

   [3. 4.] -> Option B

4. Quick Check:

   solve(A,b) output = [3. 4.] [OK]

Hint: Trust solve() output for exact numeric solutions. [OK]

Common Mistakes:

• Mixing up order of solution vector
• Trying to solve manually without decimals
• Assuming integer solutions only

4. What is wrong with this code snippet?

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)

medium

A. The code will run and print the solution without issues.

B. Vector b has wrong shape and causes a syntax error.

C. The import statement is incorrect.

D. Matrix A is singular (not invertible), so solve() will raise an error.

Solution

1. Step 1: Check matrix A properties

   Matrix A = [[1, 2], [2, 4]] has rows that are multiples (second row = 2 * first row), so it is singular (determinant = 0).

2. Step 2: Understand solve() behavior with singular matrices

   Since A is singular, it cannot be inverted, so solve() will raise a LinAlgError indicating the matrix is singular.

3. Final Answer:

   Matrix A is singular (not invertible), so solve() will raise an error. -> Option D

4. Quick Check:

   Singular matrix causes error in solve() [OK]

Hint: Check if matrix rows are multiples before solving. [OK]

Common Mistakes:

• Ignoring singular matrix errors
• Assuming solve() works for all matrices
• Confusing vector shape errors with matrix singularity

5. You have a system of 3 equations with 3 unknowns:

A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]
b = [11, 7, 28]

Using scipy.linalg.solve, what is the solution vector x?

hard

A. [1. 2. 3.]

B. [0. 1. 2.]

C. [2. 3. 1.]

D. [3. 1. 2.]

Solution

1. Step 1: Write the system of equations

   The system is:
   1x + 2y + 3z = 11
   0x + 1y + 4z = 7
   5x + 6y + 0z = 28

2. Step 2: Solve the system using substitution

   From second equation: y = 7 - 4z
   Substitute into first: x + 2(7 - 4z) + 3z = 11 -> x + 14 - 8z + 3z = 11 -> x - 5z = -3 -> x = -3 + 5z
   Substitute into third: 5(-3 + 5z) + 6(7 - 4z) = 28 -> -15 + 25z + 42 - 24z = 28 -> z + 27 = 28 -> z = 1
   y = 7 - 4*1 = 3
   x = -3 + 5*1 = 2
   Thus, x = [2. 3. 1.]

3. Final Answer:

   [2. 3. 1.] -> Option C

4. Quick Check:

   solve(A,b) = [2. 3. 1.] [OK]

Hint: Use solve() to avoid manual errors in 3x3 systems. [OK]

Common Mistakes:

• Manual calculation errors with substitution
• Mixing variable order in solution vector
• Ignoring code output for complex systems