How to prove: (-1)A = -A ?

[paso...@my-deja.com]

How can I prove that (-1)A = -A by just using vector space properties?
I don't know how I should proceed...

Question 2. Why is it relevant to prove things like this?? I think this
is most irritating.

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[Stephen Montgomery-Smith]

paso...@my-deja.com wrote:
>
> How can I prove that (-1)A = -A by just using vector space properties?
> I don't know how I should proceed...
>

First prove that 0 A = 0;

use (1 + 0) A = 1 A
use distributive law on left hand side:
1 A + 0 A = 1 A
add -(1 A) to both sides to get
0 A = 0.

Now to get what you want start with
(1 + (-1)) A = 0 A = 0
and apply similar tricks.

> Question 2. Why is it relevant to prove things like this?? I think this
> is most irritating.

It tests your ability to derive properties from axioms. It is
sometimes helpful to realise what the minimal set of axioms
are. Also, these exercises sharpen your mind.

--
Stephen Montgomery-Smith
ste...@math.missouri.edu
http://www.math.missouri.edu/~stephen

[paso...@my-deja.com]

In article <3988E724...@math.missouri.edu>,
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:

> Now to get what you want start with
> (1 + (-1)) A = 0 A = 0
> and apply similar tricks.

Would this do it:

(1 + (-1)) A = 0

1A + (-1)A = 0
A + (-1)A = 0
-A + A + (-1)A = -A
(-1)A = -A

???

I still think that this is pretty odd... hmm.. but after all, I am not
a mathematician... ;-)

[Wilbert Dijkhof]

paso...@my-deja.com wrote:
>
> In article <3988E724...@math.missouri.edu>,
> Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
>
> > Now to get what you want start with
> > (1 + (-1)) A = 0 A = 0
> > and apply similar tricks.
>
> Would this do it:
>
> (1 + (-1)) A = 0
> 1A + (-1)A = 0
> A + (-1)A = 0
> -A + A + (-1)A = -A
> (-1)A = -A
>
> ???

Indeed. Remember by definition -A is the number for which
-A + A = A + (-A) = 0.

Note that you used the associativity property in the last
line, -A + (A + (-1)A) = (-A + A) + (-1)A = (-1)A.

> I still think that this is pretty odd... hmm.. but after all, I am not
> a mathematician... ;-)
>
> Sent via Deja.com http://www.deja.com/
> Before you buy.

Wilbert

[Bill Dubuque]

paso...@my-deja.com wrote:
>
> How can I prove that (-1)A = -A by just using vector space properties?

Hint: evaluate in two different ways the following expression, by
undistributing xy+xz -> x(y+z) the overlined and underlined terms.

---------
a(-1) + a(1) + -a
-----------

This is b = -1 in the "law of signs" proof that ab = (-a)(-b)

----------------
a b + a(-b) + (-a)(-b)
------------

in my prior post http://www.deja.com/getdoc.xp?AN=555311741&fmt=text

Note in particular the dependence on the distributive law.

-Bill Dubuque