Group theory stuff

[bold flame]

All I’ve managed to notice is that the matrix $M_{\sigma}$ is just the permutation $\sigma$ applied on the basis vectors - but I’m having trouble proving that composition of permutations and matrix multiplication are the same thing (idk how to start)

[topaz ledge]

Ravi ≥ ^• ∧ •^ ≤

[stuck cape]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[wind lark]

Well you observed that $M_\sigma e_i=e_{\sigma(i)}$, hence what is $(M_\sigma M_\tau)e_i$

[topaz ledge]

Omegabet_

[bold flame]

$e_{\tau(\sigma(i))}$

[topaz ledge]

Ravi ≥ ^• ∧ •^ ≤

[wind lark]

No

you apply M_t first

[bold flame]

Oh wait

[wind lark]

then M_o

[bold flame]

I did it backwards

Sorry

$e_{\sigma(\tau(i))}$

[wind lark]

but yes $(M_\sigma M_\tau)e_i=e_{\sigma(\tau(i))}$

[topaz ledge]

Omegabet_

Ravi ≥ ^• ∧ •^ ≤

[bold flame]

It can’t be that simple right

[wind lark]

it is

[bold flame]

💀

[wind lark]

that's the same as $e_{(\sigma\tau)(i)}=M_{\sigma\tau}e_i$

[topaz ledge]

Omegabet_

[bold flame]

Associativity

Well that actually makes sense somehow

[wind lark]

so the matrices have the same action on the basis vectors, hence they are the same

[bold flame]

I could have concluded using that?? 😭😭😭

[wind lark]

given a fixed basis, yes

[bold flame]

Ah- is the basis fixed in this problem? I haven’t done linear algebra from a rigorous point of view yet

[wind lark]

yes, you just take the standard basis

[bold flame]

Oh- alr lol

Thanks for your help!

+close

[kindred sphinx]

Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

[kindred sphinx]

Thank you for your feedback! Omegabet_ has been awarded 1 . They now have 849 . They have 3 daily left for today.