Exercise 1: Independence via the Determinant Shortcut — Possible Solution
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GIVEN
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a = [3, 1]
b = [6, 2]

STEP 1: THE DETERMINANT TEST
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det([a, b]) = (3)(2) - (1)(6) = 6 - 6 = 0

Since the determinant is exactly 0, per this chapter's own Chapter 7
shortcut, a and b are NOT linearly independent - they're dependent.

STEP 2: EXPRESSING b AS A SCALAR MULTIPLE OF a
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Checking whether b = k * a for some scalar k:
  3k = 6  ->  k = 2
  1k = 2  ->  k = 2

Both components agree on k = 2, confirming b = 2a exactly.

STEP 3: WHAT THEIR SPAN ACTUALLY IS
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Since b is just a scaled copy of a (pointing along the exact same
line through the origin), b contributes no new direction. Per this
chapter's own span table, the span of {a, b} is a single LINE through
the origin - the line in the direction of a (and b) - not the whole
2D plane.

WHY THIS WORKS AS AN ANSWER
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The determinant test is applied directly using this chapter's own
formula, the dependency is confirmed independently by solving for an
exact scalar relationship between the two vectors, and the span
conclusion follows directly from this chapter's own stated span
table entry for two parallel vectors, rather than being asserted
without justification.