The Gram-Schmidt Process · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Cleaning a basis until it shines

Mathematics · Linear Algebra · ages 20-21
Name ______________________   Date ____________

  1. Apply Gram-Schmidt to (1, 0), (1, 1). What is the second orthogonal vector?

    • (0, 1)
    • (1, 0)
    • (1, 1)
    • (0, 0)

  2. Apply Gram-Schmidt to (1, 0) and (1, 1). What is the second orthogonal vector before normalising?

    • (1, 0)
    • (0, 1)
    • (1, 1)

  3. After each Gram-Schmidt step, the span of the handled vectors matches the original span so far.

    Circle one:   True   False

  4. Normalise (6, 8) to unit length. Type its first entry.

    Answer: ______________

  5. At every stage of Gram-Schmidt, what is preserved?

    • The span of the vectors processed so far
    • The length of every input vector
    • The angle between the original vectors
    • The coordinates of the input vectors

  6. After cleaning, you dot two of the finished vectors and get a nonzero value. What does that tell you?

    • The dot product is broken
    • An arithmetic slip happened, since finished vectors must be perpendicular
    • Nonzero dots are the goal of the process

  7. You swap the input order of two independent vectors. What changes in the process?

    • The intermediate steps, but the final span is the same
    • Nothing at all changes
    • The final span becomes a different space

  8. Normalize (6, 8) to unit length. What is its first entry?

    Answer: ______________

  9. A student skips the subtraction and only normalises each raw vector. Which property is lost?

    • Unit length
    • Orthogonality between the vectors
    • The number of vectors

  10. A = QR with Q orthonormal and R = [[2, 5], [0, 3]]. What is |det A|?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Cleaning a basis until it shines W1-mt_nkp231EA_E-s1

  1. (0, 1) · Subtract the projection of (1, 1) onto (1, 0), which is (1, 0), leaving (0, 1).
  2. (0, 1) · Subtract the projection (1, 0) from (1, 1) to leave (0, 1).
  3. True · Cleanup adds orthogonality but never changes the space being spanned.
  4. 0.6 · The length is 10, so dividing gives 0.6 and 0.8.
  5. The span of the vectors processed so far · Each new vector is the old one minus combinations of earlier ones, so the span never changes.
  6. An arithmetic slip happened, since finished vectors must be perpendicular · Orthonormal output means every pair dots to zero, so nonzero exposes a slip.
  7. The intermediate steps, but the final span is the same · Order steers which projections get subtracted first, not the space you end with.
  8. 0.6 · Length is 10, so divide: (0.6, 0.8).
  9. Orthogonality between the vectors · Scaling alone never removes the lean of one vector on another.
  10. 6 · |det Q| = 1, so |det A| = |det R| = 2*3 = 6.
Worksheet · LightMySky