# Thread: Understanding a linear algebra proof

1. ## Understanding a linear algebra proof

Hello!
I need a little help understanding this proof. I'm new to this and I'm a little slow =/ I've added questions in italics next to the two places I'm unsure about.

Let F be a field. There is an element 0 $\displaystyle \epsilon$ F such that a + 0 = a for all a $\displaystyle \epsilon$ F. Assume that z $\displaystyle \epsilon$ F is such that a + z = a for all a $\displaystyle \epsilon$ F. Show that z = 0.

Solution:
Suppose z $\displaystyle \epsilon$ F satisfies a + z = a for all a $\displaystyle \epsilon$ F.
Then in particular 0 + z = 0. $\displaystyle \leftarrow$ so we're simply using a + z = a from the previous line and putting a = 0 here to get 0 + z = 0?
By the commutative law for addition 0 + z = z + 0.
As a + 0 = a for all a $\displaystyle \epsilon$ F by definition of 0 we moreover have z + 0 = z. $\displaystyle \leftarrow$ I don't see how z + 0 = z can follow from a + 0 = a...I thought z + 0 = z was simply one of the axioms of a field (the neutral element for addition). I.e., I don't see how we have used 'a + 0 = a' at all in this proof.
Combining everything we get 0 = 0 + z = z + 0 = z as required.

Thank you!

2. ## Re: Understanding a linear algebra proof

Yes, since a + z = a for all a in F, we have a = 0 in particular that yields 0 + z = 0.

Similar situation occuring in your second question. The axiom is a + 0 = a for all a in F. So in particualr when a = z, we see that z + 0 = z. a is any arbitrary element and z is a specific unknown element (which we soon shall see is zero).

I hope this helps.