Decide which of the following functions on R² are inner products and which are not

Question

5.1. Decide which of the following functions on R² are inner products and which are not. For x = (x1, x2), y = (y1, y2) in R2 (1) (x, y) = x1y1x2y2, (2) (x, y) = 4x1y1 +4x2y2 – x1y2 – x2y1, (3) (x,y) = x192 − x291, (4) (x, y) = x1y1 + 3x2y2, (5) (x, y) = x1y1 − x1y2 − x2y1 + 3x2y2. Exercises 5.1 Inner products are (2), (4), (5). 5.2 For the last condition of the definition, note that (A, A) = tr(ATA) = Σi₁ja² = 0 if and only if a¡j = 0 for all i, j. 5.4 (1) k 3. = (5.5) (3) || ƒ || = ||g|| = √1/2, The angle is 0 if n = m, if n ‡m. 5.6 Use the Cauchy-Schwarz inequality and Problem 5.2 with x = (a₁,…, an) and y = (1, …, 1) in (R”, .). (5.7) (1) 37/4, 19/3. (2) If (h, g) = h(+2+ c) = 0 with h 0 a constant and g(x) = ax² +bx+c, then (a, b, c) is on the plane + + c = 0 in R³.

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