Statement Let V be any vector space and let x, y, z be in V. Prove that if x + z = y +z, then x = y. Proof piem Suppose V is any vector space and x, y, z are any vectors in V. Suppose x + z = y + z. Show that x = y Since z is in V, there exists a vector -z in V such that z + (-z) = 0 Thus, x + z = y + z (x+z) + (-z) = (y+z) + (-z) x + (z + (−z)) = y + (z+ (−z)) because x + 0 = y +0 because and thus x = y Therefore, x + z = y + z implies x = y, and V has the cancellation property. Allowed answers because addition is commutative implies and thus by properties of the zero vector. V is a vector space, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition

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