9.1 P1

Let V be a vector space. Prove that the zero element is unique.

Proof: Suppose x & y are zero elements of V. That is for every 
v in V, v + x = v, and v + y =v. Then

                 x = x + y = y + x = y.

Thus, x & y are the same element of V. This proves the claim. 

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9.1 P4 (I will use 0 for the zero vector and 0.0 for the 
real number zero.)

Let V be a vector space, with v in V and r a real number.
Prove that if rv = 0, then either r = 0.0 or v = 0.

Proof: Assume rv=0 and suppose r is not 0.0. Then we can multiply 
both sides by 1/r.

    (1/r)(rv) = (1/r)0 
    (1/r*r)v  = (1/r)0  (axiom 9)
       1v     = (1/r)0  (definition of the real number 1/r)
        v     = (1/r)0  (Axiom 10)
        v     =    0    (Theorem 9.1.2b)

Thus, either r=0.0 or v=0. 

[It is possible for both r=0.0 and v=0 to be true. The word "or" 
does not exclude this.]  

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