Proof using axioms for a field

In summary, Halmos' book "Finite Dimensional Vector Spaces" is proving to be difficult for the novice mathematician. The problem in question is proving that (-1)a = -a. The mathematician is having difficulty getting past this point after trying for a day.
  • #1
msbell1
25
0
Hi, I am trying to work through Finite Dimensional Vector Spaces by Halmos, and I am having some difficulty with the first problem on page two (the specific problem is included below). The last class I took involving formal proofs was linear algebra about 8 years ago, and I am very rusty, but I am trying to regain some of those skills now. Here is the problem:

Homework Statement


Prove that (-a)(-b)=ab (assuming that F is a field and that a, b, and c belong to F)

Homework Equations


A reminder of the axioms:
addition is defined in the usual way, and it is commutative (a + b = b + a), associative (a + (b + c) = (a + b) + c), 0 is the unique additive identity such that a + 0 = a, and -a is the unique additive inverse such that a + -a = 0
multiplication is defined in the usual way:
it is commutative (ab = ba), associative (a(bc) = (ab)c), 1 is the unique additive identity such that a1 = a, and 1/a is the unique multiplicative inverse such that a(1/a) = 1.


The Attempt at a Solution


In the previous part of this problem I was asked to prove that (-1)a = -a, so I will use that proposition to prove the next part. I start out by using that to write:

(-a)(-b)=(-1)a(-1)b

However, once I write this I am stuck--for the past day I have been thinking about this off and on and have not been able to make any progress, and I was wondering if anyone could suggest how I should proceed.

Thank you for the help!
 
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  • #2
Do you know what (-1)(-1) equals? If not could you prove it equals 1?
 
  • #3
Thanks for the reply!
I forgot to include that multiplication is distributive with respect to addition:
a(b+c) = ab + ac (which I will use later)

Would it be ok to first rewrite (-1)a(-1)b as (-1)(-1)ab and then start working separately on the (-1)(-1) part?

like continuing the proof as

(-a)(-b) = (-1)(-1)ab

Then forgetting about a and b and working on (-1)(-1):

(1 + (-1))(-1) = 0 (1 + -1 = 0 (-1 is unique additive inverse of 1 such that 1 + -1 = 0) and I already had to prove that 0a = a0 = 0)

(-1)(1 + (-1)) = 0 (by commutativity of multiplication)

(-1)(1) + (-1)(-1) = 0 (multiplication is distributive w.r.t. addition)

-1 + (-1)(-1) = 0 (because 1 is multiplicative identity)
1 + -1 + (-1)(-1) = 1 + 0
(1 + -1) + (-1)(-1) = 1 + 0 (because addition is associative)
0 + (-1)(-1) = 1 + 0 (because 1 + -1 = 0)
(-1)(-1) + 0 = 1 + 0 (because addition is commutative)
(-1)(-1) = 1 + 0 (because a + 0 = a)
(-1)(-1) = 1 (because a + 0 = a)

Then plugging this result into

(-a)(-b) = (-1)(-1)ab
(-a)(-b) = 1ab
(-a)(-b) = ab

Is that an ok proof? How would a mathematician prove this? Thanks again for your reply.
 
  • #4
yup looks good, so an outline would look like:

lemma 1: -a = (-1)a
lemma 2: (-1)(-1)=1
proof:
(-a)(-b) =(-1)a(-1)b
(-1)a(-1)b = (-1)(-1)ab
(-1)(-1)ab = 1ab
1ab = ab
 
  • #5
Thanks a lot--especially for the outline!
 

1. What are axioms for a field?

Axioms for a field are a set of rules or properties that define the fundamental operations and properties of a mathematical field. They serve as the building blocks for the development of mathematical proofs in a field.

2. How do axioms for a field differ from axioms in other mathematical systems?

Axioms for a field are specific to the properties of a field, which is a mathematical structure defined by two operations (addition and multiplication) and their corresponding properties. Other mathematical systems may have different sets of axioms that define their fundamental operations and properties.

3. What is the purpose of using axioms in mathematical proofs?

The use of axioms in mathematical proofs ensures that the conclusions drawn are based on a consistent and logical set of rules. They provide a solid foundation for mathematical reasoning and allow for the development of new theorems and concepts.

4. Can axioms for a field be proven?

No, axioms for a field cannot be proven. They are accepted as true and serve as the starting point for mathematical proofs. However, their consistency and logical coherence can be verified through the use of mathematical proofs.

5. How are axioms used to prove theorems in a field?

Axioms for a field are used as the starting point for mathematical proofs, where logical deductions are made to arrive at a conclusion. Theorems in a field are proven by applying these axioms and other previously proven theorems in a systematic and logical manner.

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