Use the properties of vector addition and scalar multiplication from the following theorem. Properties of Vector Addition and Scalar Multiplication in R" Let u, v, and w be vectors in R", and let c and d be scalars. 1. u + v is a vector in R". 2. u + v = v + u Closure under addition Commutative property of addition Associative property of addition Additive identity property Additive inverse property Closure under scalar multiplication Distributive property Distributive property Associative property of multiplication Multiplicative identity property 3. (u + v) + w = u + (v + w) 4. u + 0 = u 5. u + (-u) = 0 6. cu is a vector in R". 7. cu + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1(u) = u Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 distributive property -(-v) + (-v) = v + (-v) transitive property of equality -(-v) + (-v) + v = v + (-v) + v associative property of addition -(-v) + ((-v) + v) = v + ((-v) + v) distributive property |-(-v) + 0 = v + 0 distributive property -(-v) = v multiplicative identity property X X

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