Show That There Is Only One Linear Transformation Proof

[mmmboh]

Show That There Is Only One Linear Transformation Proof Help Please!

Hi, I have been trying this problem for a couple of days, I have done a proof but I don't know if it makes sense. If you want I can scan it and show it, but if someone can show me how to do it that would be more than amazing, I have midterms coming up soon

Thanks.

 

[Discover85]

Haha... I'm going to assume you are in my class because this is on my assignment for this week!

I had difficulty with it too but I found this which was immensely helpful.

Scroll down to proposition 8.3. In the proof they talk about proposition 4.1, which we proved on a previous assignment.

Hope that helps :)

 

[mmmboh]

Hey thanks a lot that cleared things up!

 

[HallsofIvy]

Suppse there were two such transformations, T_1 and T_2 T_1\ne T_2. Any vector v can be written in terms of the basis vectors, v= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n.

Then T_1(v)=T_1(a_1v_1+ a_2v_2+\cdot\cdot\cdot+ a_nv_n)= T_1(a_1v_1)+ T_1(a_2v_2)+ \cdot\cdot\cdot+ T_1(a_nv_n)a_1T_1(v_1)+ a_2T_1(v_2)+ \cdot\cdot\cdot+ a_nT_1(v_n)= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n.

Also T_2(v)= T_2(a_1v_1+ a_2v_2+\cdot\cdot\cdot+ a_nv_n)= T_2(a_1v_1)+ T_2(a_2v_2)+\cdot\cdot\cdot+ T_2(a_nv_n)= a_1T_1(v_1)+ a_2T_2(v_2)+ \cdot\cdot\cdot+ a_nT_2(v_n)= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n.

That is, for any vector, v, T_1(v)= T_2(v) contradicting the assumption that T_1\ne T_2.