Solving linear systems (solve) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the function that solves linear systems.

SciPy

from scipy.linalg import [1]

A = [[3, 1], [1, 2]]
b = [9, 8]
x = [1](A, b)
print(x)

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

Asolve

Binv

Cdet

Deig

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'inv' instead of 'solve' to solve the system.

Trying to use 'det' or 'eig' which are unrelated to solving linear systems.

The solve function from scipy.linalg is used to solve linear systems Ax = b.

2fill in blank

medium

Complete the code to solve the system and print the solution vector.

SciPy

from scipy.linalg import solve

A = [[4, 2], [3, 1]]
b = [10, 8]
x = solve(A, [1])
print(x)

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

A[10, 8]

Bb

CA

Dsolve

Attempts:

3 left

💡 Hint

Common Mistakes

Passing matrix A as second argument instead of vector b.

Passing the function name 'solve' as argument.

The second argument to solve is the vector b representing the right side of the equation Ax = b.

3fill in blank

hard

Fix the error in the code to correctly solve the linear system.

SciPy

from scipy.linalg import solve

A = [[1, 2], [3, 4]]
b = [5, 6]
x = solve([1], b)
print(x)

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

AA

Bb

C[5, 6]

Dsolve

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping the order of arguments to solve.

Passing the vector b as first argument.

The first argument to solve must be the coefficient matrix A.

4fill in blank

hard

Fill both blanks to create a dictionary of solutions for each variable from the system.

SciPy

from scipy.linalg import solve

A = [[2, 1], [1, 3]]
b = [8, 13]
x = solve(A, b)
solution = { [1]: x[0], [2]: x[1] }
print(solution)

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

A"x1"

B"x2"

C"a"

D"b"

Attempts:

3 left

💡 Hint

Common Mistakes

Using matrix or vector names as keys instead of variable names.

Using unquoted keys causing syntax errors.

The dictionary keys should be variable names like 'x1' and 'x2' to label the solution components.

5fill in blank

hard

Fill the blanks to create a dictionary comprehension that maps variable names to their solutions if the solution is positive.

SciPy

from scipy.linalg import solve

A = [[1, -1], [2, 3]]
b = [1, 12]
x = solve(A, b)
result = { [1]: x[i] for i in range(2) if x[i] [2] 0 }
print(result)

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

A"x" + str(i+1)

B>

C>=

D"var" + str(i)

Attempts:

3 left

💡 Hint

Common Mistakes

Using incorrect string concatenation for keys.

Using '<' or '<=' instead of '>' in the condition.

The dictionary comprehension creates keys like 'x1', 'x2' for variables. The condition checks if the solution is greater than zero.

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