quiz.json

{
  "questions": [
    {
      "stage": "pre",
      "question": "What is an eigenvector of a matrix?",
      "options": ["The largest row in the matrix", "A vector that the matrix only scales (never rotates) when multiplied", "A vector perpendicular to all columns of the matrix", "The diagonal entries of the matrix expressed as a vector"],
      "correct": 1,
      "explanation": "An eigenvector v satisfies Av = lambda*v, meaning the matrix A only stretches v by the scalar factor lambda (the eigenvalue) without changing its direction."
    },
    {
      "stage": "pre",
      "question": "What does the determinant of a 2D transformation matrix represent geometrically?",
      "options": ["The angle of rotation applied by the matrix", "The factor by which the matrix scales area", "The number of eigenvectors the matrix has", "The trace of the matrix"],
      "correct": 1,
      "explanation": "The determinant measures how much the transformation scales area. det=1 preserves area (rotation), det=2 doubles area, det=0 crushes to a lower dimension, and det=-1 preserves area but flips orientation."
    },
    {
      "stage": "post",
      "question": "Why does the order of matrix transformations matter? (i.e., why is R @ S different from S @ R?)",
      "options": ["Matrix addition is not commutative", "Matrix multiplication is not commutative: rotating then scaling gives a different result than scaling then rotating", "The determinants are different for each order", "One order produces a larger matrix than the other"],
      "correct": 1,
      "explanation": "Matrix multiplication is not commutative. Rotating (1,0) by 90 degrees then scaling by (2,0.5) gives (0,0.5), but scaling first then rotating gives (0,2). The geometric operations compose differently."
    },
    {
      "stage": "post",
      "question": "In a recurrent neural network, what happens when the weight matrix has eigenvalues with magnitude greater than 1?",
      "options": ["The network learns faster", "Outputs explode exponentially over time steps (exploding gradient problem)", "The network becomes more stable", "The eigenvalues converge to 1 over training"],
      "correct": 1,
      "explanation": "Repeated multiplication by a matrix amplifies the eigenvalue directions. Eigenvalues > 1 cause exponential growth (exploding gradients), while eigenvalues < 1 cause exponential decay (vanishing gradients)."
    },
    {
      "stage": "post",
      "question": "The matrix A = [[2, 1], [1, 2]] has eigenvalues 3 and 1. What does eigendecomposition A = V @ D @ V^(-1) reveal?",
      "options": ["A is equivalent to two rotations", "A stretches space by 3x along the [1,1] direction and leaves the [1,-1] direction unchanged", "A compresses all vectors by a factor of 2", "A has rank 1 and maps all vectors to a line"],
      "correct": 1,
      "explanation": "The eigenvalue 3 with eigenvector [1,1] means A stretches 3x along the diagonal. The eigenvalue 1 with eigenvector [1,-1] means A leaves the anti-diagonal unchanged. D holds {3,1}, V holds the eigenvectors."
    }
  ]
}