Question

Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): -1 1 ( 2 5 3 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, g(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2x2 matrices: (You'd decided what the inner product was on a previous math practice. Use your inner product again here). {{ : %][1 :] [: :][{ :) 1 1 1

Answer 1

361 A collection of basis & h, de..., daß is orthogonal if < Vish; >= 0 tisj & {1,2...m3, & itj Further, the basis is orthogmal if (1) holds additionally, Juill=< Vi , U = 1 ti= 1,2,..., w and ya 3 = 0 - O (3)(-1) + 0) (2) + (1) (3) īram > 1 (5) < > = (-0(1) + (2)(5) + (3) (-3) = 0 > () > = (0(3) + 50) + (-3)(O) = 0 < 5 - 3 is or Therefore, the given basis Now, (3) (3) thogonal. <(:). (3)> = (318) + (10)+ ** (C) = 10 1)I = No # 1 is not basis Hence, the or thonormal. vector space is An inner product given given by - pa) - do + 9x + ₂x² - 9(x) = botbat b. x² <p (x), q (x) > = P(-1) 9(-) + (6) 9 (0) + P(1) 9 (1) Therefore, For f(x), f(0) = – 3. f(-1) = -2 = f (1) For g(x), 96) = 9(-1) = 9(1) = 4 <f(x), 9 (2) >= (-3) (4) + (-2) (4) + (-2) (4) = - 2870 Therefore, the basis is not orthogonal, hence not orthogonal, hence not orthonormal also

We don't need to check for all vectors because even if for a pair of vector, if the condition doesn't hold then the basis is not orthogonal.