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(Section 5.1)
20 points
1. a. Let V be the set of real-valued functions that are defined at each x in the
interval (-oo, co). If f = f(x) and g = g(x) are two functions in V and if
k is any scalar, then define the operations of addition and scalar
multiplication by
(f+ 9) = f(x)+g(x)
kif = kf(x)
Verify the Vector Space Axioms for the given set of vectors.
b. Let V consist of the form u = (u1, u2, ..., Un, ...) in which
U1, U2, ..., Un; ... is an infinite sequence of real numbers. Define two
infinite sequences to be equal if their corresponding components are equal,
and define addition and scalar multiplication componentwise by
utv = (U1, U2, ..., Un, ...) + (21, 02, ..., Un; ...)
= (U1 + v1, U2 + V2, ..., Un + Un, ...)
ku = (kul, kuz, ..., kun, ...).
Prove that the given set is a vector space.

Here the first can be solved by F=f(x) and G(x) =g(x) (F+G) =F+G =&gt;F(x)... View the full answer

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