Additive identity over linear transformation

[autre]

Given vector spaces V, W over a field, and linear transformation T:V\rightarrow W, prove T(0_{v})=0_{w} where 0_v and 0_w are additive identities of V and W.

I'm trying to use the definition of additive identity. So, \forall\vec{v}\in V,\vec{v}+0=\vec{v+0=0}. Where do I go from here?

 

[CompuChip]

I don't really see why v + 0 = 0... unless v = 0.
However, you may want to look at v + (-v) = 0.

 

[autre]

Woops, I meant \forall\vec{v}\in V,\vec{v}+0=\vec{0+v=\vec{v}}. Not the additive inverse.

 

[CompuChip]

Yep, I guess you can use that as well.
You will still need the additive inverse at some point though (if not in V, then at least in W to show that T(v) - T(v) = 0_W).

 

[Theorem.]

Here's a hint: Consider T(0_v+0_v).

 

[autre]

Would this line of reasoning work?

T(0_{v})=T(0_{v}+0_{v})=T(0_{v}+\vec{v}-\vec{v}+0_{v})=T(0_{v}+\vec{v})-T(0_{v}+\vec{v})=T(\vec{v})-T(\vec{v})=\vec{w}-\vec{w}=0_{w}.

 

[Theorem.]

That looks good to me! In fact you can make it even simpler If you want by noting T(0_v)=T(0_v+0_v)=T(0_v) + T(0_v) which is only possible if T(0_v)=?

 

[autre]

T(0_{v})=0_{w}, or the additive identity in W?

 

[Theorem.]

Exactly , it must be the zero of W.

 

[Fredrik]

Would this line of reasoning work?

T(0_{v})=T(0_{v}+0_{v})=T(0_{v}+\vec{v}-\vec{v}+0_{v})=T(0_{v}+\vec{v})-T(0_{v}+\vec{v})=T(\vec{v})-T(\vec{v})=\vec{w}-\vec{w}=0_{w}.

It's correct, but you're using the following two results without proving them

• T(-x) is the additive inverse of T(x) for all x.
• -0=0

The first one is no more obvious than the statement you're trying to prove.

If you're going to use these results, there's no need to involve a new vector v. You could just write T(0)=T(0+0)=T(0+(-0))=T(0)+T(-0)=T(0)-T(0)=0. I prefer the method Theorem suggested.

 

[CompuChip]

An alternative way to write it down is:
T(0) = T(0 + 0) = T(0) + T(0) because of linearity.
Now add the additive inverse of T(0) (whatever it is, but there must be one) to both sides, and you get
T(0) + (-T(0)) = T(0) + T(0) + (-T(0))
0 = T(0)