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.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top