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 $T_1\ne T_2$.