Question

1) Show that if V is a vector space, then -(-v)=v for all v V.

2) The empty set is not a vector space. It fails to satisfy only one of the requirements from the definition. Which one?

Answer 1

To show that -(-v) = v for all v ∈ V, we need to use the properties of vector addition and scalar multiplication.

Step-by-step explanation:

To show that -(-v) = v for all v ∈ V, we need to use the properties of vector addition and scalar multiplication.

1. Let v be an arbitrary vector in V. Then, we have:
   • -(-v) = (-1)(-v) (Using the property of scalar multiplication)
   • = (-1)(-1)(v) (Using the property of scalar multiplication)
   • = 1(v) (Using the property of scalar multiplication)
   • = v (Using the property of scalar multiplication)
   Therefore, -(-v) = v for all v ∈ V.