LU decomposition in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the LU decomposition function from scipy.

SciPy

from scipy.linalg import [1]

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

Aeig

Bsvd

Cinv

Dlu

Attempts:

3 left

💡 Hint

Common Mistakes

Importing unrelated functions like svd or eig.

Using functions from other modules.

The lu function from scipy.linalg performs LU decomposition.

2fill in blank

medium

Complete the code to perform LU decomposition on matrix A.

SciPy

P, L, U = [1](A)

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

Alu

Bsvd

Cinv

Deig

Attempts:

3 left

💡 Hint

Common Mistakes

Using svd or eig instead of lu.

Trying to call lu without arguments.

The lu function returns the permutation matrix P, lower triangular matrix L, and upper triangular matrix U.

3fill in blank

hard

Fix the error in the code to correctly get the lower triangular matrix L from LU decomposition.

SciPy

_, [1], _ = lu(A)

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

AU

BL

CP

DQ

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning U or P to the variable for L.

Using an undefined variable name.

The second returned value from lu is the lower triangular matrix L.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each row index to the sum of the corresponding row in matrix U.

SciPy

row_sums = {i: [1] for i in range(U.shape[0]) if [2] > 0}

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

Asum(U[i])

Bi

DU[i]

Attempts:

3 left

💡 Hint

Common Mistakes

Using the entire row U[i] as a key.

Using the sum as the key instead of the value.

The comprehension sums each row U[i] and uses the row index i as the key.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps each column index to the maximum value in that column of matrix L, but only if the max is greater than 0.

SciPy

col_max = {j: max(L[:, [1]]) for j in range(L.shape[[2]]) if max(L[:, j]) [3] 0}

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

Aj

B1

C>

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using row index instead of column index.

Using incorrect comparison operators.

The comprehension iterates over columns j, gets the max of column j in L, and includes it if the max is greater than 0.

Practice

1. What does LU decomposition do to a square matrix A?

easy

A. Splits A into three matrices: P, L, and U

B. Calculates the determinant of A

C. Finds the inverse of A directly

D. Transforms A into a diagonal matrix only

Solution

1. Step 1: Understand LU decomposition purpose

   LU decomposition breaks a matrix into three matrices: a permutation matrix P, a lower triangular matrix L, and an upper triangular matrix U.

2. Step 2: Compare options with definition

   Only Splits A into three matrices: P, L, and U correctly describes this splitting. Other options describe different matrix operations.

3. Final Answer:

   Splits A into three matrices: P, L, and U -> Option A

4. Quick Check:

   LU decomposition = P, L, U matrices [OK]

Hint: LU always produces P, L, U matrices from A [OK]

Common Mistakes:

• Confusing LU with determinant calculation
• Thinking LU gives inverse directly
• Assuming LU produces only diagonal matrix

2. Which of the following is the correct way to import the LU decomposition function from SciPy?

easy

A. import lu from scipy.linalg

B. import scipy.lu

C. from scipy import lu_decompose

D. from scipy.linalg import lu

Solution

1. Step 1: Recall SciPy LU import syntax

   The LU decomposition function is in the scipy.linalg module and imported as lu.

2. Step 2: Check each option's syntax

   from scipy.linalg import lu uses correct Python import syntax. Options B, C, and D are invalid or incorrect module/function names.

3. Final Answer:

   from scipy.linalg import lu -> Option D

4. Quick Check:

   Correct import = from scipy.linalg import lu [OK]

Hint: Use 'from scipy.linalg import lu' to import LU [OK]

Common Mistakes:

• Trying to import lu directly from scipy
• Using wrong function name like lu_decompose
• Incorrect import syntax like 'import lu from ...'

3. What is the output of this code?

import numpy as np
from scipy.linalg import lu
A = np.array([[4, 3], [6, 3]])
P, L, U = lu(A)
print(np.round(L, 2))

medium

A. [[1. 0. ]
[0.67 1. ]]

B. [[4. 3. ]
[6. 3. ]]

C. [[1.5 0. ]
[0. 1. ]]

D. [[0.67 1. ]
[1. 0. ]]

Solution

1. Step 1: Perform LU decomposition on matrix A

   Using scipy.linalg.lu on A = [[4,3],[6,3]] gives matrices P, L, U where L is lower triangular with 1s on diagonal and multipliers below.

2. Step 2: Calculate L matrix values

   L = [[1, 0], [6/4, 1]] = [[1, 0], [1.5, 1]]. But scipy.linalg.lu applies row permutations, so actual L is [[1,0],[0.67,1]] after permutation.

3. Final Answer:

   [[1. 0. ] [0.67 1. ]] -> Option A

4. Quick Check:

   L matrix lower triangular with 1 diagonal = [[1,0],[0.67,1]] [OK]

Hint: L is lower triangular with 1s on diagonal [OK]

Common Mistakes:

• Confusing L with original matrix A
• Ignoring row permutations affecting L
• Rounding errors in L matrix values

4. Identify the error in this LU decomposition code:

import numpy as np
from scipy.linalg import lu
A = np.array([[1, 2], [3, 4]])
P, L, U = lu(A)
print(L * U)

medium

A. Function lu returns only two matrices, not three

B. Matrix multiplication should use @, not *

C. LU decomposition requires square matrix, but A is not square

D. Numpy array A must be float type for LU

Solution

1. Step 1: Understand matrix multiplication in numpy

   The code uses L * U, which does element-wise multiplication, not matrix multiplication.

2. Step 2: Correct multiplication operator

   Matrix multiplication requires L @ U or np.dot(L, U) to multiply matrices properly.

3. Final Answer:

   Matrix multiplication should use @, not * -> Option B

4. Quick Check:

   Use @ for matrix multiply, not * [OK]

Hint: Use @ for matrix multiply, * is element-wise [OK]

Common Mistakes:

• Using * instead of @ for matrix multiplication
• Thinking LU returns only two matrices
• Assuming matrix must be float type always

5. You want to solve the system Ax = b using LU decomposition with SciPy. Which steps correctly describe the process?

hard

A. Decompose A into L, U only; solve Lx = b; solve Ux = b

B. Decompose A into P, L, U; solve Ux = Pb; solve Ly = x

C. Decompose A into P, L, U; solve Ly = Pb; solve Ux = y

D. Decompose A into P, L, U; solve Px = b; solve Ly = x; solve Uz = y

Solution

1. Step 1: Perform LU decomposition with permutation

   LU decomposition gives P, L, U such that PA = LU.

2. Step 2: Solve system using forward and backward substitution

   First solve Ly = Pb (forward substitution), then solve Ux = y (backward substitution) to find x.

3. Final Answer:

   Decompose A into P, L, U; solve Ly = Pb; solve Ux = y -> Option C

4. Quick Check:

   Use P, L, U and solve Ly=Pb then Ux=y [OK]

Hint: Solve Ly=Pb then Ux=y after LU [OK]

Common Mistakes:

• Ignoring permutation matrix P in solving
• Solving equations in wrong order
• Confusing variables in substitution steps