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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3 Inner Product Consider the inner product defined by (x, y) = xy = xy + 2y2, x, y € R²x1 (the scalar I [3] and b = [2¹] verify product of the two vectors; also called dot product). For a = 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 2 (x, y) = x¹ y = 2x1y1 F192 - 1291 +31292- 3 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.
Transcribed Image Text:3 Inner Product Consider the inner product defined by (x, y) = xy = xy + 2y2, x, y € R²x1 (the scalar I [3] and b = [2¹] verify product of the two vectors; also called dot product). For a = 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 2 (x, y) = x¹ y = 2x1y1 F192 - 1291 +31292- 3 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.
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