QR decomposition in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

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

SciPy

from scipy.linalg import [1]

# Now you can use qr() to decompose matrices

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

Aqr

Bsvd

Ceig

Dinv

Attempts:

3 left

💡 Hint

Common Mistakes

Importing the wrong function like svd or eig.

Trying to import from numpy.linalg instead of scipy.linalg.

The qr function from scipy.linalg performs QR decomposition.

2fill in blank

medium

Complete the code to perform QR decomposition on matrix A.

SciPy

Q, R = qr([1])

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Aqr

BQ

CA

DR

Attempts:

3 left

💡 Hint

Common Mistakes

Passing Q or R instead of A to the qr function.

Passing the function name qr instead of the matrix.

You pass the matrix A to the qr function to get matrices Q and R.

3fill in blank

hard

Fix the error in the code to correctly compute the QR decomposition and print Q.

SciPy

from scipy.linalg import qr

A = [[1, 2], [3, 4]]
Q, R = qr([1])
print(Q)

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AA

B[1, 2], [3, 4]

Cqr

DQ

Attempts:

3 left

💡 Hint

Common Mistakes

Passing multiple lists instead of a single matrix variable.

Passing the function name or Q instead of the matrix.

The variable A must be passed as a single argument to qr. Passing the list of lists directly without brackets causes an error.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each column index to the norm of that column in Q.

SciPy

col_norms = {i: [1] for i in range(Q.shape[[2]])}

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Anp.linalg.norm(Q[:, i])

B0

C1

DQ.shape[0]

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong axis index for shape.

Calculating norm on rows instead of columns.

The dictionary comprehension calculates the norm of each column i in Q. The number of columns is Q.shape[1].

5fill in blank

hard

Complete the code to create a dictionary comprehension that maps each row index to the sum of absolute values in that row of R, but only for rows where the sum is greater than 1.

SciPy

row_sums = {i: sum(abs(R[i, :]) ) for i in range(R.shape[[1]]) if sum(abs(R[i, :])) > 1}

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

A:]

B0

C:

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Using incorrect slice notation like ':]' or '1'.

Using the wrong axis index for shape.

The code sums the absolute values of all columns in each row i of R. The row count is R.shape[0]. The slice : selects all columns.

Practice

1. What does QR decomposition do to a matrix A?

easy

A. Converts A into a lower triangular matrix only

B. Splits A into two diagonal matrices

C. Splits A into an orthogonal matrix Q and an upper triangular matrix R

D. Finds the inverse of A directly

Solution

1. Step 1: Understand QR decomposition definition

   QR decomposition splits a matrix into two parts: an orthogonal matrix Q and an upper triangular matrix R.

2. Step 2: Compare options with definition

   Only Splits A into an orthogonal matrix Q and an upper triangular matrix R correctly describes this split. Others describe different matrix operations.

3. Final Answer:

   Splits A into an orthogonal matrix Q and an upper triangular matrix R -> Option C

4. Quick Check:

   QR decomposition = Q and R matrices [OK]

Hint: QR always produces Q (orthogonal) and R (upper triangular) [OK]

Common Mistakes:

• Confusing QR with LU decomposition
• Thinking QR produces diagonal matrices
• Assuming QR finds matrix inverse

2. Which of the following is the correct way to import and use QR decomposition from scipy?

easy

A. from scipy.linalg import qr
Q, R = qr(A)

B. import scipy.qr
Q, R = scipy.qr(A)

C. from scipy import qr
Q, R = qr(A)

D. import qr from scipy.linalg
Q, R = qr(A)

Solution

1. Step 1: Recall scipy QR import syntax

   The correct import is from scipy.linalg import qr.

2. Step 2: Check usage correctness

   Calling qr(A) returns Q and R. from scipy.linalg import qr Q, R = qr(A) matches this exactly.

3. Final Answer:

   from scipy.linalg import qr Q, R = qr(A) -> Option A

4. Quick Check:

   Correct import and call = from scipy.linalg import qr Q, R = qr(A) [OK]

Hint: Use 'from scipy.linalg import qr' to access QR function [OK]

Common Mistakes:

• Trying to import qr directly from scipy
• Using incorrect import syntax
• Calling qr without importing

3. Given the matrix A = [[1, 2], [3, 4]], what is the shape of Q and R after Q, R = qr(A)?

medium

A. Q is (2, 2) and R is (2, 1)

B. Q is (1, 2) and R is (2, 2)

C. Q is (2, 1) and R is (1, 2)

D. Q is (2, 2) and R is (2, 2)

Solution

1. Step 1: Check input matrix shape

   Matrix A has shape (2, 2).

2. Step 2: Understand QR output shapes

   For a square matrix, Q and R both have the same shape as A, which is (2, 2).

3. Final Answer:

   Q is (2, 2) and R is (2, 2) -> Option D

4. Quick Check:

   Square matrix QR outputs same shape matrices [OK]

Hint: Q and R shapes match input matrix shape for square matrices [OK]

Common Mistakes:

• Assuming R has fewer columns
• Confusing Q shape with R shape
• Thinking Q or R is 1D

4. What is wrong with this code snippet for QR decomposition?

from scipy.linalg import qr
A = [[1, 2], [3, 4]]
Q, R = qr(A)
print(Q)

medium

A. Nothing is wrong with this code snippet

B. The qr function does not return two values

C. Matrix A should be a numpy array, not a list of lists

D. Import statement is incorrect

Solution

1. Step 1: Verify import statement

   The import from scipy.linalg import qr is correct.

2. Step 2: Check qr function returns

   qr(A) returns two values Q and R.

3. Step 3: Validate input type

   List of lists for A works as qr accepts array-like inputs and converts internally.

4. Step 4: Check print statement

   print(Q) correctly outputs the Q matrix.

5. Final Answer:

   Nothing is wrong with this code snippet -> Option A

6. Quick Check:

   Code runs successfully, produces correct Q and R [OK]

Hint: scipy.linalg.qr accepts lists of lists as array-like input [OK]

Common Mistakes:

• Thinking Python lists cannot be passed to qr
• Misunderstanding qr return values
• Incorrect import statements

5. You want to solve the system Ax = b using QR decomposition where A is a 3x3 matrix and b is a vector. Which step correctly describes how to find x after computing Q, R = qr(A)?

hard

A. Compute x = Q @ b then solve Ry = x

B. Compute y = Q.T @ b then solve Rx = y using back substitution

C. Compute y = R @ b then solve Qx = y

D. Compute x = R.T @ Q @ b

Solution

1. Step 1: Use QR decomposition to rewrite Ax = b

   Since A = QR, rewrite as QRx = b.

2. Step 2: Multiply both sides by Q.T

   Multiply by Q.T to get Rx = Q.T b. Define y = Q.T b.

3. Step 3: Solve upper triangular system

   Now solve Rx = y using back substitution to find x.

4. Final Answer:

   Compute y = Q.T @ b then solve Rx = y using back substitution -> Option B

5. Quick Check:

   Use Q.T b and solve Rx = y [OK]

Hint: Multiply b by Q.T, then solve Rx = y [OK]

Common Mistakes:

• Multiplying b by Q instead of Q.T
• Trying to solve Qx = y instead of Rx = y
• Ignoring that R is upper triangular