Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v, w) = 0, ||u|| = 1, ||v|| = v3, ||w|| = 2. %3D Show that u +v = w.

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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a)
Consider the following definition of inner product:
Definition of Inner Product Space
Suppose u, v, and w are all vectors in a vector space Vand c is any scalar. An inner product on
the vector space V is a function that associates with each pair of vectors in V, say u and v, a real
number denoted by (u, v) that satisfies the following axioms:
1. (u, v) = (v, u)
2. (u + v, w) = (u, w) + (v, w)
3. (cu, v) = c{u, v)
4. (u, u) > 0 and (u, u) = 0 if and only if u = 0
Suppose that u, v, and w are vectors in an inner product space such that
(u, v) = 1,
(u, w) = 5,
(v, w) = 0,
||u|| = 1,
||v|| = v3,
||w|| = 2.
Show that u +v = w.
Transcribed Image Text:a) Consider the following definition of inner product: Definition of Inner Product Space Suppose u, v, and w are all vectors in a vector space Vand c is any scalar. An inner product on the vector space V is a function that associates with each pair of vectors in V, say u and v, a real number denoted by (u, v) that satisfies the following axioms: 1. (u, v) = (v, u) 2. (u + v, w) = (u, w) + (v, w) 3. (cu, v) = c{u, v) 4. (u, u) > 0 and (u, u) = 0 if and only if u = 0 Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v, w) = 0, ||u|| = 1, ||v|| = v3, ||w|| = 2. Show that u +v = w.
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