Why is my proof incorrect? - Lin alg

[wispy mirage]

👋 Welcome to your Help Thread!

Hey @ebon coral, Thanks for sharing your math problem with us. While you wait for a Helper to help you, we want to share some vital information with you.

​

● Please take a moment to read the helpee guidelines. This will make sure that your post follows the helpee rules of our community.

​

● Please don't ping <@&775784618955505685>, <@&1283689826742440016>, <@&819616364188139550>, or <@&624327278137966593> for help because their job is to take care of the server's administrative tasks, not to answer queries directly. However, if you have a problem with how a Helper is acting, you can ping a Helper Moderator.

​

● It's always very useful if you can show us the work you've done so far. This makes it easier for our Helpers to find mistakes and help you get to the right answer.

Once you've got your answer and you're all set, you can close this thread by typing +close, and then you can click the Thank buttons to show your Helper some appreciation.

[ebon coral]

I was told it was incorrect but I dont quite understand why

I'm not a mathematician so my rigor might be a little off (notation etc)

also dont mind pinging me when anyone resopnds

[unborn flare]

How do you know f_0 exists?

Each line is also just off, you also never showed phi was linear

[lofty bane]

what does c + lambda*epsilon mean, what does small number greater than 0 mean? who ever said K = real numbers?

[ebon coral]

I proved phi is linear earlier (not on the doc), i'm purely confused about the surjectivity

[lofty bane]

your task is to show phi is a linear surjection

[ebon coral]

Would you two say that my proof lacks rigor or that the logic to my solution is just totally wrong?
In other words, could tiny changes like more grounded definitions etc save this?

[unborn flare]

You've also said W is finite

[unborn flare]

I mean, as aL for some reason pointed out here... epsilon isn't a thing

If w is a vector in W, can you find a linear map that sends v to that specific w

[lofty bane]

let f,g in Hom (V,W), does it follow that phi (f+g) = phi (f) + phi (g)?@ebon coral

[unborn flare]

If you're going to interrupt, at least read the chat.

[unborn flare]

.

[ebon coral]

phi(f+g) would just be (f+g)(v) right? And since f and g are linear it follows that (f+g)(v) = f(v) + g(v), or did I go wrong there

[ebon coral]

right, since epsilon is a number not a vector? Would it work if I defined epislon to be a basis vector multiplied by such an infentisemaly small scalar?

[unborn flare]

Epsilon isn't a thing period

[lofty bane]

ok I suppose you have proved linearity elsewhere

now let w in W be arbitrary. Find a suitable linear map f : V to W such that f(v) = w. If w = 0, you can take f = 0.

[ebon coral]

I am using epislon such like an epsilon analogous to epislons you'd see in analysis, is that not allowed in lin alg?

[unborn flare]

You're not

[ebon coral]

what if I say there exsits an epsilon > 0 , would that make it valid

[unborn flare]

Epsilon in analysis is just an arbitrary positive number

It'd make it better, but you've not assumed anything about the field

[lofty bane]

what does positive mean? K is just any ol' field

[unborn flare]

So you don't know > exists

[ebon coral]

Right that makes sense

So is my method hopeless? (I have a proper solution to this problem btw using bassisses, I just wanted to know where I want wrong and if it was salvagable)

[lofty bane]

it is hopeless

[unborn flare]

Yes

[lofty bane]

stick to the definition of surjectivity

[hallow fulcrum]

(f+g)(v)=f(v)+g(v) is true in general for functions. they don't need to be linear.

[unborn flare]

They're already being helped.

[wispy mirage]

@ebon coral

:HelpIcon:| Help Reminder

Hello .kobe., this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.

[ebon coral]

+close

[wispy mirage]

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.

[wispy mirage]

Thank you for your feedback! aL has been awarded 1 . They now have 968 . They have 3 daily left for today.

[wispy mirage]

Thank you for your feedback! Omega has been awarded 1 . They now have 898 . They have 3 daily left for today.