SOLUTION: Let x, y, and z be vectors in a vector space V. Prove that if: x + y = x + z then y = z.

Question 846536: Let x, y, and z be vectors in a vector space V.
Prove that if:
x + y = x + z
then y = z.
Found 2 solutions by Fombitz, rothauserc:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
In the vector space there exists a vector xi such that when added to vector x, you will get the zero vector.
Add xi to both sides.
x+y+xi=x+z+xi
Since addition is commutative,
x+xi+y=x+xi+z
0+y=0+z
By identity,
y=z
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
use additive inverse of x which is -x, then
x -x + y = x -x + z
0 + y = 0 + z
use additive identity, 0 + y = y, 0 + z = z, therefore
y = z

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