35  Exercises — Linear Algebra Track

Three exercises to consolidate the concepts from the four chapters of the linear algebra track.

35.1 Exercise 1 — Flatten and compare

Create two 4 \times 4 synthetic images — a horizontal gradient (constant within each row, increasing left-to-right) and a vertical gradient (constant within each column, increasing top-to-bottom). Flatten both to vectors.

  1. Compute their dot product and cosine similarity.
  2. Are they orthogonal?
  3. Subtract the mean from each, then re-check. Are the mean-subtracted versions orthogonal?

Hint: a horizontal gradient has the same value in each column; a vertical one has the same value in each row.

Expected: the raw dot product is non-zero, but the mean-subtracted versions are orthogonal.

35.2 Exercise 2 — Energy change under transforms

Take a 3 \times 3 checkerboard patch (alternating black and white). Apply these transforms and compute the energy (\|\mathbf x\|^2) of each:

  1. Multiply by 3 (contrast increase)
  2. Add 100 (brightness increase)
  3. Rotate the flattened vector using a random orthogonal matrix — Q, _ = np.linalg.qr(np.random.randn(9, 9))

Which transforms preserve energy?

Expected: only the rotation preserves energy.

35.3 Exercise 3 — Decompose a real-ish image

Create an 8 \times 8 synthetic image with both a pattern (diagonal stripe) and a brightness (overall mean around 150):

image = np.where(np.abs(np.subtract.outer(np.arange(8), np.arange(8))) < 2,
                 255.0, 50.0)
  1. Decompose it into brightness + pattern components using mean subtraction.
  2. Verify the two components are orthogonal.
  3. Add a brightness offset of +80 and show the pattern component is unchanged.

Expected: identical pattern components before and after the brightness offset; the orthogonality holds in both cases.