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msbell1
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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:
Prove that (-a)(-b)=ab (assuming that F is a field and that a, b, and c belong to F)
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.
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!
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!