MHB Cameron V 's question at Yahoo Answers (Equality of linear maps)

  • Thread starter Thread starter Fernando Revilla
  • Start date Start date
  • Tags Tags
    Linear
AI Thread Summary
The discussion revolves around proving that if two linear transformations L1 and L2 yield the same outputs for a basis of a vector space V, then L1 must equal L2. The proof begins by establishing that if L1 equals L2, then their outputs for each basis vector are trivially equal. Conversely, if L1 and L2 produce the same results for all basis vectors, a generic vector x in V can be expressed as a linear combination of these basis vectors. By applying the linear transformations to this combination, it is shown that L1(x) equals L2(x), leading to the conclusion that L1 equals L2. The thread encourages further questions to be posted in a dedicated math help forum.
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
Here is the question:

Prove:
for each i = 1, 2, ... n
than L1(vi) = L2(vi)
only if L1 = L2

I can this of the concept in my head and I think I understand it but I am having trouble actually putting the proof on paper. Any help is appreciated.
Thanks

Here is a link to the question:

{v1, v2, ... , vn} is a basis for V. L1 and L2 are two linear transformations mapping V into a vectorspace W.? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello Cameron V,

If $L_1=L_2$, trivially $L_1(v_i)=L_2(v_i)$ for all $i=1,\ldots,n$. On the other hand if $L_1(v_i)=L_2(v_i)$ for all $i=1,\ldots,n$, choose a generic $x\in V$. As $\{v_1,\ldots,v_n\}$ is a basis of $V$, $x=\alpha_1+\ldots +\alpha_n v_n$ for some scalars $\alpha_1,\ldots,\alpha_n$. Then, for all $x\in V$: $$\begin{aligned}L_1(x)&=L_1(\alpha_1v_1+\ldots +\alpha_n v_n)\\&=\alpha_1L_1(v_1)+\ldots +\alpha_n L_1(v_n)\\&=\alpha_1L_2(v_1)+\ldots +\alpha_n L_2(v_n)\\&=L_2(\alpha_1v_1+\ldots +\alpha_n v_n)\\&=L_2(x)\\&\Rightarrow L_1=L_2\end{aligned}$$ If you have further questions, you can post them in the http://www.mathhelpboards.com/f14/ section.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top