how can one check that a set with two operations, addition and scalar multiplication, is a vector space? what must a zero vector satisfy? how about the negative of a vector?

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 17EQ
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d.2 lineal algebra 

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how can one check that a set with two operations, addition and scalar multiplication, is a vector space?
what must a zero vector satisfy? how about the negative of a vector?
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