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Math Help - Understanding a linear algebra proof

  1. #1
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    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 \epsilon F such that a + 0 = a for all a \epsilon F. Assume that z \epsilon F is such that a + z = a for all a \epsilon F. Show that z = 0.

    Solution:
    Suppose z \epsilon F satisfies a + z = a for all a \epsilon F.
    Then in particular 0 + z = 0. \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 \epsilon F by definition of 0 we moreover have z + 0 = z. \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!
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  2. #2
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    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.
    Thanks from chlovee
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