Consider the inner product defined by (x, y) = xy = y + x2y2, x, y € R²x1 (the scalar [3] [2¹] verify product of the two vectors; also called dot product). For a = and b = the following (in)equalities by calculating the LHS (left-hand side) and the RHS (right-hand side) separately and comparing them:¹ (a) the Cauchy-Schwarz inequality (a, b)² ≤ (a, a). (b, b); (b) the triangle inequality ||a+b||≤||a|| + ||b||; (c) the parallelogram law ||a+b||2+ ||ab||22||a||²+2||b||². = Repeat the above exercise for the inner product defined by (x, y) = x² 27 [²₁₁3¹]v=20₁1 y = 2x191 192-12/1 + 3x292. 2 Note that this is indeed an inner product as the matrix is positively-definite. Remember that the norm has to be calculated based on the inner-product used.

Consider the inner product defined by (x, y) = xy = y + x2y2, x, y € R²x1 (the scalar [3] [2¹] verify product of the two vectors; also called dot product). For a = and b = the following (in)equalities by calculating the LHS (left-hand side) and the RHS (right-hand side) separately and comparing them:¹ (a) the Cauchy-Schwarz inequality (a, b)² ≤ (a, a). (b, b); (b) the triangle inequality ||a+b||≤||a|| + ||b||; (c) the parallelogram law ||a+b||2+ ||ab||22||a||²+2||b||². = Repeat the above exercise for the inner product defined by (x, y) = x² 27 [²₁₁3¹]v=20₁1 y = 2x191 192-12/1 + 3x292. 2 Note that this is indeed an inner product as the matrix is positively-definite. Remember that the norm has to be calculated based on the inner-product used.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.1: Length And Dot Product In R^n

Problem 83E: Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars...

See similar textbooks