Proof involving vector spaces and linear transformations

[killpoppop]

1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v)



2. -T(v) denotes the additive inverse of T(v)



3. I think i'm really overcomplicating it =/ But i have

0v = T( v - v ) = T(v) + T(-v)

Then add -T(v)

0v + -T(v) = T(v) + T(-v)

(T(v) + T(-v)) -T(v) = T(v) + T(-v)

then

T(-v) = T(v) + T(-v)

then i suppose it could go to

T(-v) = 0v

but that doesn't help i'm going round in circles. Basically i need a starting point.

[Fredrik]

Looks more like you need someone to tell you when to stop. :smile:

When you obtain 0=T(v)+T(-v), you're done. If T(-v) has that property, it is the additive inverse of T(v). You should however explain more clearly why T(v+(-v))=0.

Hm, are you allowed to use that 0x=0 for all x without proof? If not, it's easy to prove using the trick 0x=(0+0)x.

[killpoppop]

Cheers!

Right so when i get:

0v = T(v) + T(-v)

i just move the T(v) over because 0v=0?

giving

-T(v) = T(-v)

seems a bit too easy! I've already proved 0x=0 .

Can you explain more about

'You should however explain more clearly why T(v+(-v))=0.'

please =]

[NoMoreExams]

You shouldn't need to move anything over, you know that 0v = 0, therefore at the step where you have T(v) + T(-v) = 0 that means T(-v) is the additive inverse of T(v), remember what definition of additive inverse is!

[Fredrik]

Can you explain more about

'You should however explain more clearly why T(v+(-v))=0.'

I'm not sure if I can tell you anything new without giving you the complete solution. What you want to prove is that T(v)+T(-v)=0. And the proof of that goes like this:

T(v)+T(-v)=T(v+(-v))=...=0

Can you fill in the missing steps?