How do I prove linear independency or dependency here

[signal bridge]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well: https://www.patreon.com/mathsdiscord

[cunning frigate]

If there's only the the trivial solution a = b = c = 0, then the functions are independent.

[mossy sable]

yea i know that part

[cunning frigate]

Though, if you remember the definition of sinh(x), you can easily instantly say the answer.

[mossy sable]

but I dont know how to solve

[mossy sable]

like whats next

[cunning frigate]

Now group terms with the same exponent.

[cunning frigate]

Yes.
Now, you need the left part to be zero. What system of equations do you get?

[mossy sable]

like this?

[cunning frigate]

Yes.

[mossy sable]

what approach could I use to solve the system

[cunning frigate]

Gauss, as usual.

[mossy sable]

sure

gimme a sec

[cunning frigate]

Though, at this point it's pretty much obvious what the solution looks like.

[mossy sable]

yeah ig

[mossy sable]

unique solution (the trivial one)?

so independent?

[cunning frigate]

No, of course not.

Clearly, there are infinitely many solutions.

[mossy sable]

ah

[cunning frigate]

You can't have a homogeneous system with more variables than equations have only the trivial solution.

[mossy sable]

true

I thought of it the opposite

since all rows have pivots

but thats not how it works

cuz not all columns have

[mossy sable]

so infinite solutions, which means they are dependant?

[cunning frigate]

Yeah!

But now let's solve this exercise in a waaaay shorter way.

[mossy sable]

cool

ty

[cunning frigate]

Suppose the elements are u = sinh(x), v = e^x, w = e^(-x).
We know that sinh(x) = (1/2)(e^x - e^(-x)). So:
u = (1/2)v - (1/2)w
Thus, u, v and w are linearly dependent.

[mossy sable]

oh

lol

thats smart

[cunning frigate]

That's what I meant in the beginning 😄

[mossy sable]

well ty man

im doing my linear algebra homework so might open another ticket in a while if I get stuck

[cunning frigate]

You're welcome!

Oh, sure!

[mossy sable]

+ty

how was it

[mossy sable]

ty

[red needle]

@mossy sable has given 1 rep to @cunning frigate