I've been trying to solve the last one all day with no success

How do get the last part

assume that
d(N1/N) > d(N2/N)

since they are both finite, N1/N must have at least one more dimension than N2/N. so d(V/(N1/N)) < d(V/(N2/N)), which implies d(V/N1) < d(V/N2)

>so d(V/(N1/N)) < d(V/(N2/N))
How did that follow from what you wrote previously? What kind of object is this to begin with? As far as I can tell quotient spaces are defined as A/B where B is a subspace of A.

V/N, V/N1, V/N2, N1/N, N2/N are all finite,

d(N/N1) > d(N/N2), so d(V/(N/N1)) < d(V/(N/N2)) since N/N1 is a larger subspace than N/N2.

>V/N, V/N1, V/N2, N1/N, N2/N are all finite,

* finite dimensional

>(V/(N/N1)
N/N1 isn't a subspace of V

meant to write N1/N and N2/N, not N/N1, N/N2, sorry.

>N1/N and N2/N
not subspaces of V either

why not? maybe i'm not understanding what / does.

elements of N1/N are usually written as the equivalence class [n1] or n1+N where n1 is in N1

see
en.wikipedia.org/wiki/Quotient_space_(linear_algebra)#Definition

huh, i dunno then.

I figured it out. Let + denote direct sum.
V=N_1+M_1
V=N_2+M_2
where d(M_1)=d(M_2) and both exist. This is given by the hypothesis.
Now N_i/N is a subspace of V/N, and V/N is finite dimensional by the second question, since V/N_i is. Therefore d(N_i/N) exists for both i, and we have N_i=N+D_i, where D_i is finite dimensional.
Then we have V=N+M_i+D_i. Now we note that in general if V=R+Q=R+S, and both Q and S are finite dimensional, then they have the same dimension. This is because V=R+Q=R+S implies V/R is isomorphic to Q and S.
So we have d(M_1$D_1)=d(M_2$D_2) where $ now denotes set addition. but since M_i$D_i = M_i+D_i, we have d(M_1) + d(D_1) = d(M_2) + d(D_2)
so that d(D_1) = d(D_2). Therefore d(N_1/N) = d(N_2/N)

>N_i/N is a subspace of V/N

how could this be true if it weren't also a subspace of V? isn't V/N a subspace of V itself?

You make it so complicated looking, but maybe that is the way to state what I've drawn.
Or maybe it isn't!

V/N is a completely different structure from V, so no they arent subspaces. V/N is the set of all sets of the form v + N, so all translations of N translated along an element of V. It is necessary that N be a subset of V for the translation to make sense. Now consider all the translates of N using elements of V, and then consider all the translates of N using N_i, the latter is N_i/N, and this is clearly a subset of V/N, because every element of N_i used to translate N is also in V so is a member of V/N. With the operations of set addition and set multiplication by scalars "scaling up" all elements in the set, V/N becomes a vector space and similarly this subset becomes a subspace because it contains the equivalent of a 0 element and has closure under the operations.

>how could this be true if it weren't also a subspace of V?
let c be a scalar and n_a+N, n_b+N elements of N_i+N (so n_a and n_b are elements of N_i)

then
c*(n_a + N) + (n_b + N)
= (c*n_a+n_b)+N
is an element of N_i/N since c*n_a+n_b is an element of N_i (N_i is a subspace of V by definition)

therefore N_i/N is a subspace of V/N

>isn't V/N a subspace of V itself?
no

i'll have to read more about it, never seen quotient spaces before

thanks, that's much clearer.

come on user, it's in the name
take V's basis, find the basis of a subspace N_i in V's basis, and remove those vectors. count them and you have the codimension

>come on user, it's in the name
i think i get it now, but in my defense, this is probably the most misleading sentence ever written on wikipedia
>In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero

it's not really that misleading, N is the zero vector in the quotient V/N

I'm the one who made the thread and even I didnt understand what the hell that was supposed to mean

>i'm the one who made the thread and even i
what do you mean "even i"? you have no repute. if anything, your decision to make a hw thread for such a simple question gives you negative repute.

If it was so simple why did you do nothing to assist with it. Your explanations post the solution were also unhelpful to other people asking questions. If you read the thread you would realize that I understand what codimension means, especially since I solved the problem. You claim that that the wikipedia's intrpoductory statement is clear, but it is not meant to be rigorous statement, which is why they use quotation marks with "collapse." Theres only 4 IP's here, so I take it you are the same person repeatedly telling the other how simple the concept of codimension is, despite them only learning the definition within the same hour. Your comments have been completely useless.

i am not the only one replying to you, someone else is. i'm the one who posted the picture to help you visualize it, which i think was helpful and if it wasn't, you didn't say anything.
also
>a statement can't be clear if it isn't rigorous
>"i understand codimension but i don't understand what that statement means"
>"i understand codimension, especially since i solved that problem"
codimension is literally just the difference between the dimension of a vector space and its subspace (in the finite case; it's slightly more nuanced otherwise)
i don't understand why you're upset about this, undergraduate math is mostly studying simple things, and i was calling you out for assuming undeserved repute. even if all of the stuff you just wrote meant anything, it still wouldn't change those facts. don't take it personally, what i said was factual, i'm sorry it offended you

Who cares if I said "even I" you're the one taking it personally and going on about I have "no repute" to claim something is confusing or not. If that picture is anything to show for it you are not the person whose opinion should be noted if something is clear or not. I didnt respond to it because it made no sense

>can't read set diagrams
wew

and who cares? it made you seem like you were another pretentious undergrad here and i enjoy knocking them down a peg; i care
i've got nothing to take personally, you're getting all flustered and i'm telling you there's nothing to get flustered about.

why did my diagram make no sense? i'm sorry you didn't find it helpful

Saying I'm a pretentious undergrad is the sign you took it personally, it wasnt already obvious. If this "simpleton undergraduate" can explain this simple concept better than you can its you who needs to go down a peg.

>sign that i took it personally
no it isn't, i just picked the proper adjective to describe you and judging by the rest of your new post, i'm even more sure of it
>simpleton
nobody called you a simpleton, i was saying all undergraduates study simple stuff for the most part; that's nothing to be ashamed about, that's how it is and how we all started. you have to learn to write rigorously and think clearly about the simple stuff before you can move on. again, nothing to be flustered about
>explain this simple concept better than you
you still haven't told me specifically what's wrong - i answered your question with a diagram, you answered it with the language of math, and they're both saying the same thing.

it's okay to make a mistake so long acknowledge it, no need to argue for the sake of arguing