1. ## Vector Help

I dont understand how to do this. the question is

If a and b are vectors who that the equation a + x = b has the only solution that x = b - a

any help?

2. I certanly could not understand what the problem is.

Could you please possibly explain what is the problem statement?

3. Originally Posted by p0oint
I certanly could not understand what the problem is.

Could you please possibly explain what is the problem statement?
sorry i wrote it wrong

If a and b are vectors, show that the equation a + x = b has the unique solution x = b - a

4. The result follows from the definition of a vector space. Use the fact that inverses are unique, associativity of addition, and the like, to prove it.

5. Originally Posted by Ackbeet
The result follows from the definition of a vector space. Use the fact that inverses are unique, associativity of addition, and the like, to prove it.
i haven't learned that yet though...we have only gone past the principle of linear independence.

6. Hmm. Start out as follows:

$\displaystyle a+x=b$ Given.
There is a vector $\displaystyle 0$ such that for any vector $\displaystyle y$, we have $\displaystyle y+0=0+y=y$.
Since $\displaystyle a$ is a vector, there is another vector, call it $\displaystyle -a$, such that $\displaystyle a+(-a)=0$. [EDIT Here]

We add this vector $\displaystyle -a$ to both sides: $\displaystyle -a+(a+x)=-a+b$.
Since addition is associative and commutative, we may rewrite as follows:
$\displaystyle (-a+a)+x=b-a$. Can you finish?

7. Originally Posted by Ackbeet
Hmm. Start out as follows:

$\displaystyle a+x=b$ Given.
There is a vector $\displaystyle 0$ such that for any vector $\displaystyle y$, we have $\displaystyle y+0=0+y=y$.
Since $\displaystyle a$ is a vector, there is another vector, call it $\displaystyle -a$, such that $\displaystyle a+(-1)=0$. We add this vector $\displaystyle -a$ to both sides: $\displaystyle -a+(a+x)=-a+b$.
Since addition is associative and commutative, we may rewrite as follows:
$\displaystyle (-a+a)+x=b-a$. Can you finish?
oh i think i get it. then since a - a = O (as in the zero vector) we get

O + x = b - a
x = b - a

but how do we show that it is a unique solution? is that assumed?

btw how do you get this
$\displaystyle a+(-1)=0$

8. The usual method to prove something is unique is to assume there is another solution, and prove it is equal to the one you already found. So, assume there is another solution, $\displaystyle \hat{x}$, and show that $\displaystyle \hat{x}=x$. I think you'll need the uniqueness of the zero vector, and the uniqueness of inverses to show the result.

9. Originally Posted by Ackbeet
The usual method to prove something is unique is to assume there is another solution, and prove it is equal to the one you already found. So, assume there is another solution, $\displaystyle \hat{x}$, and show that $\displaystyle \hat{x}=x$. I think you'll need the uniqueness of the zero vector, and the uniqueness of inverses to show the result.
oh allright so i dont think we will need that yet. thanks.

10. Good catch. I've fixed that error. Hopefully it makes more sense now.