Guided Proof Prove that a nonempty set W is asubspace of a

Chapter 4, Problem 4.3.51

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Guided Proof Prove that a nonempty set W is asubspace of a vector space V if and only if ax + by isan element of W for all scalars a and b and all vectorsx and y in W. Getting Started: In one direction, assume W is asubspace, and show by using closure axioms thatax + by is an element of W. In the other direction,assume ax + by is an element of W for all scalars aand b and all vectors x and y in W, and verify that W isclosed under addition and scalar multiplication.(i) If W is a subspace of V, then use scalar multiplicationclosure to show that ax and by are in W. Now useadditive closure to get the desired result.(ii) Conversely, assume ax + by is in W. By cleverlyassigning specific values to a and b, show that W isclosed under addition and scalar multiplication.

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