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Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•,•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x,x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (iv) (v) (x, ay) = a(x, y) for all x, y V (x, y+z) = (x, y) + (x, z) for all (x, y) (y,x) for all x, y € V. = (p, q) = xεν. n and for all Example: Verify that each of the following mappings define a real inner product. i=1 x, y, z ε ν. (d) Let x₁,x2,...n be distinct real numbers and let (,): PnX Pn → R be defined as p(x₁)q(xi), a EF. for all p, q Pn.

Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•,•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x,x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (iv) (v) (x, ay) = a(x, y) for all x, y V (x, y+z) = (x, y) + (x, z) for all (x, y) (y,x) for all x, y € V. = (p, q) = xεν. n and for all Example: Verify that each of the following mappings define a real inner product. i=1 x, y, z ε ν. (d) Let x₁,x2,...n be distinct real numbers and let (,): PnX Pn → R be defined as p(x₁)q(xi), a EF. for all p, q Pn.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (p, q) = 72 xεν. Example: Verify that each of the following mappings define a real inner product. (d) Let x1,x2,... En be distinct real numbers and let ( P(xi)q(xi), i=1 and for all x,y,z € V. a EF. ): Pn x Pn → R be defined as for all p, q E Pn.
Transcribed Image Text:Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (p, q) = 72 xεν. Example: Verify that each of the following mappings define a real inner product. (d) Let x1,x2,... En be distinct real numbers and let ( P(xi)q(xi), i=1 and for all x,y,z € V. a EF. ): Pn x Pn → R be defined as for all p, q E Pn.
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