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