27.5. The Cauchy–Schwarz Inequality

Try these first

1. 1.

   Given vectors u = (1, 1) and v = (1, -1), calculate |⟨u,v⟩| and check if it satisfies the Cauchy-Schwarz Inequality.

   Hint

   Remember to find the norms first.

2. 2.

   For u = (2, 2) and v = (1, 3), determine the inner product and norms, checking the Cauchy-Schwarz condition.

   Hint

   Calculate each component separately.

3. 3.

   What does the Cauchy-Schwarz Inequality state about two vectors u and v?

   • |⟨u,v⟩| ≥ ∥u∥ · ∥v∥
   • |⟨u,v⟩| ≤ ∥u∥ · ∥v∥
   • |⟨u,v⟩ = ∥u∥ · ∥v∥

   Hint

   Think about how inner products relate to vector lengths.

4. 4.

   What condition must hold for equality in the Cauchy-Schwarz Inequality?

   • True
   • False

   Hint

   Consider the definition of linear dependence.

5. 5.

   Prove that for any vectors u = (u1, u2) and v = (v1, v2) in R², |u1v1 + u2v2| ≤ √(u1² + u2²) * √(v1² + v2²) holds.

   Hint

   Identify the components systematically for clarity.

6. 6.

   If a triangle's vertices correspond to vectors A, B, and C, derive the triangle inequality using the Cauchy-Schwarz Inequality.

   Hint

   Focus on the relationships each side forms with respect to the others.