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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.2: Inner Product Spaces

Problem 94E

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Consider the mapping (·, ·) : R" × R” → R defined by
n
(u,v) = |ui||vi|, Vu, v€ Rn.
i=1
Show which defining properties of inner products are satisfied by this mapping. Provide a counterex-
ample for properties not satisfied.

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Show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). ⟨u, v⟩ = u1v1 − 2u2v2 + u3v3
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Let f be a function that takes elements of R?«R? into R? as follows: f(u, v) = |u1v1|+|u2v2|, where u = (u1, U2) e R² and v = (v1, v2) E R². Show that f cannot be an inner product by showing that one of the properties is not satisfied.
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T^2 is the torus, RP^2 is the real projective plane
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(3) Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) (x, y) = (u+x+1,v+y+1) for all (u, v) and (x, y) in V. Define an operation of "scalar multiplication" by a(r, y) = (ar, ay) for all a ER and (x, y) = V. Under the operations and the set V is not a vector space. The vector space axioms (see 5.1.1 (1)-(10)) which fail to hold are and
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