If T is a linear transformation, then T(0)=0 -- OBVIOUSLY TRUE, BUT HOW SHOULD I PROVE IT?

[TheWrathOfMath]

I know that T:V-->U is a linear transformation if the following conditions are met:



Obviously for T(0)=0, I can write:
T(a*0)=aT(0)
However, how do I proceed to show that the value of T(0) is zero?

And I cannot choose v1=v2=0 in order to prove that the first condition is met, since v1 and v2 must be two different vectors in V.

Perhaps I am going about this the wrong way?

 

[TheWrathOfMath]

Never mind. I managed to prove it.

 

[Steven G]

0 = 0+0
So T(0) = T(0+0) = T(0) + T(0)

Finish up from here.

 

[Steven G]

Out of curiosity, why do you think that that theorem is obvious true? I too thought that it was obvious when I first saw it but at the same time I was thinking oh 0+0=0.

 

[pka]

I would have done it as: [imath]\bf T(1)=T(1+0)=T(1)+T(0)[/imath] the implies [imath]\bf T(0)=0[/imath].

[imath][/imath][imath][/imath]

 

[Steven G]

I would have done it as: [imath]\bf T(1)=T(1+0)=T(1)+T(0)[/imath] the implies [imath]\bf T(0)=0[/imath].

[imath][/imath][imath][/imath]

Nice, I'm glad that you showed me this.