# Vector Spaces and Scalar Multiplication

1)Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R. Prove axioms (VS1)=For all x,y, x+y=y+x (commutativity of addition), (VS3)= There exist an element in V denoted by 0 such that x+0=x for each x in V.,(VS4)= For each element x in V there exist an element y in V such that x+y=0.,(VS5)= For each element x in V, 1x=x.,(VS7)=For each element a in F and each pair of elements x,y in V, a(x+y)= ax+ay. for this space.

2)Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.

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1)Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R. Prove axioms (VS1)=For all x,y, x+y=y+x (commutativity of addition), (VS3)= There exist an element in V denoted by 0 such that ...

#### Solution Summary

Vector spaces and scalar multiplication are investigated. The elements of functions in different spaces are determined.