Exercise 1: Local-to-World Transform for an Object at (3,4) Rotated 90° — Possible Solution
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BUILDING THE TRANSFORMATION MATRIX
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Using this chapter's own formula, T = [[cos theta, -sin theta, tx],
[sin theta, cos theta, ty], [0,0,1]], with theta=90 degrees,
tx=3, ty=4:

    cos(90) = 0, sin(90) = 1

    T = [[0, -1, 3],
         [1,  0, 4],
         [0,  0, 1]]

APPLYING IT TO THE LOCAL POINT (2,0)
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As a homogeneous point, (2,0) becomes (2,0,1). Multiplying:

    row 1: 0*2 + (-1)*0 + 3*1 = 0 + 0 + 3 = 3
    row 2: 1*2 +   0*0  + 4*1 = 2 + 0 + 4 = 6
    row 3: 0*2 +   0*0  + 1*1 = 1

Result: (3, 6, 1) - the world-space position is (3, 6).

VERIFYING BY REASONING THROUGH THE TWO STEPS SEPARATELY
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This can be double-checked by doing rotation and translation as two
separate steps, per this chapter's own explanation of what the
combined matrix represents: rotating (2,0) by 90 degrees around the
origin maps (x,y) to (-y,x), giving (-0, 2) = (0,2). Adding the
translation (3,4) gives (0+3, 2+4) = (3,6) - exactly matching the
single-matrix result.

WHY THIS WORKS AS AN ANSWER
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The answer builds the matrix correctly from this chapter's own
formula, carries out the matrix-vector multiplication explicitly
rather than skipping to the answer, and cross-checks the result using
the separate rotate-then-translate reasoning as an independent
verification, matching the chapter's own emphasis on confirming
results two different ways.