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9786503 No.9786503 [Reply] [Original]

Got a task here. Gotta prove that vectors 1, sinx, cosx, sin2x, cos2x create a basis in the linear shell over these vectors (set of all the linear combinations of them).

Now it basically comes down to proving their linear independence.

What's the easiest way? Calculating Grammian seems like an overkill.

How can you prove that sin2x is independent of sinx, cosx, cos2x?

>> No.9786508

>>9786503
use trig identities

>> No.9786510

>>9786503
https://en.wikipedia.org/wiki/Wronskian

>> No.9786512

>>9786503
Plug in five different x-values and show that the determinant is nonzero.

>> No.9786514

>>9786510
this

>> No.9786522

>>9786508
then what? Say I represent sin2x as 2sinxcosx. Well now what?

What is the strict proof here?

>> No.9786540

>>9786510
Welp, the second and the fourth strings are the same if you multiply one of them by -1. So that should mean vector are dependent.

Clearly inapplicable wtf

>> No.9786561

>>9786540
The Wronskian can give false positives, this just means you need to use a more powerful test. The one you mention in the OP does not give false positives.

>> No.9786566

>>9786561
Well the fourth string is the second one multiplied by -1, so the determinant must be 0. Yet clearly these vectors are independent.

>> No.9786567

>>9786566
>Well the fourth string is the second one multiplied by -1, so the determinant must be 0. Yet clearly these vectors are independent.
Yeah... that is because, as I just said, the Wronskian can give false positives.

>> No.9786574

>>9786567
You said the one I mentioned does not give false positives. And yet it just fucking did. WTF?

>> No.9786575

>>9786503
change basis to {1, exp(x), exp(-x), exp(2x), exp(-2x)} and then do what >>9786512
said. or just do what he said.

>> No.9786578

>>9786575
Too much fucking work, shit aint supposed to be this hard trust me

>> No.9786581

>>9786574
The Grammian does not give false positives. The Wronskian does. Learn to read mate.

Anyways, if the Wronskian failed what I would do if I was you is write everything in terms of exponentials, which you can. After that create an equivalent basis in exponential functions and use any of the theorems that guarantee independence between different exponential functions.

>> No.9786584

>>9786581
Fuck Grammian, that shit is tedious as a motherfucker

There MUST BE an easier way, than the exponential hurr durr.

Is there really no way to show that sinx cannot be a linear combination of 1, cosx, sin2x, cos2x and same for those?..

>> No.9786588

>>9786503
[eqn]\sin \left ( mx \right )\cos \left ( nx \right )=\frac{\sin \left ( mx+nx \right )+\sin \left ( mx-nx \right )}{2}[/eqn]

>> No.9786590

>>9786584
Well, there is an easier way someone told you hours ago and you did not listen. Literally set up the linear system and plug in some values until you have enough equations to conclude independence.

That would be the easy caveman tier way of doing it, and it seems you are looking for that. Just fucking do it.

>> No.9786596

>>9786590
Tedious.

I actually fucked up taking derivatives.

So there ARE no linearly dependent strings or rows on the Wrongskin materics. Is that a guarantee that the vectors are independent?

>> No.9786601

>>9786596
>Tedious
It's what you would have to do if the Wronskian failed.

>Is that a guarantee that the vectors are independent?
I think you are an asshole so I'm gonna let you look it up yourself.

>> No.9786605

>>9786601
I knew people would take it wrong on the internet, I was actually just amusing myself responding like that.

Appreciate your answers anyway.

>> No.9786607

>>9786510
More like Wrongskian

>> No.9787690

>>9786605
>I was actually just pretending to be an asshole.
Very convincing job of it.

>> No.9789169

>>9786503
They are orthogonal wrt the dot product [math]\langle f, g \rangle = \int_0^{2\pi} f(x) g(x) dx [/math]; hence linear independent.

>> No.9789179

>>9789169
this

>> No.9789181

>>9789169
BOOM!

>> No.9790017

you nerds can study linear independence all you want but i quit uni to make games and now i am an independent adult

>> No.9790042

>>9790017
Needing to post about it on the internet for validation is a great sign that things are going great for you and that you definitely are not grasping at straws to keep the suicidal thoughts in.