Let R² have the inner product of Example 1. Show that the Cauchy–Schwarz inequality holds for x = .(3,– 4) and y = (–4; 3).

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EXAMPLE 1 Fix any two positive numbers-say, 4 and 5-and for vectors u = (₁, ₂) and v = (v₁, v₂) in R², set (u, v) = 4u1v₁ + 5u2v2 Show that equation (1) defines an inner product. (1)