native carbon
Hey, I'm doing my Linear Algebra hw and this is something I'm stuck on since we didn't cover this type of thing in class at all. I got these sequences in a set and need to show that V2 ⊆ V1.
I've already shown that V1 ⊆ V2 but this way around I have no idea how I would go about this. We haven't had sequences yet so I'm not sure what type of method I'd need to use to solve this. Any sort of pointers or tips would be greatly appreciated :)
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Hey, I'm doing my Linear Algebra hw and this is something I'm stuck on since we ...
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oak parrot
induction comes to mind
native carbon
yeah that's what I also initially was thinking but I couldn't come up with where to start there
oak parrot
well you have to take an educated guess for your base case/induction assumption
keep in mind that a sequence in V2 is supposedly in this form
$$ a_{n+3} - a_n = 3(a_{n+2} - a_{n+1}) $$
tight pagodaBOT
aL
native carbon
okay to show that V1 ⊆ V2 i put the definition of v1 into this and then in the end came to the conclusion that it's just an+3
is the induction proof the other way not just the same then kind of
i had to show this in the problem before. it just says that for any x0,x1,x2 you can chose the c0,c1,c2 so the equation in the bottom is true
isn't that just my induction start and then since it's for the first 0-2 i need to show n -> n+3 and then I just copy paste what I already did anyway since I can use it there as my induction assumption/requirement
but idk it doesn't feel like I can use that as my induction start because it has nothing to do with V2. Not sure if I can argue that for any sequence in V2 you need to choose the starting 3 anyway so since you can have the starting three in that polynomial form I can use it again
oak parrot
i understand V1 subset V2 is already done
native carbon
yes
oak parrot
now take a sequence in V2
by assumption it follows this recursive definition
where does your indexing start, do you consider 0 in N?
native carbon
yeah we consider 0 in N
oak parrot
ok so the smallest case is a3 = 3a2 - 3a1 + a0
native carbon
yeah
native carbon
yeah i got that
oak parrot
your base case is n=3
you need to figure out what to take as c0 c1 c2 based on that
and then make your induction assumption
native carbon
so I can't reuse this?
oak parrot
i don't know what it says
native carbon
okay so what is means is just
for any starting 3 numbers you can have a sequence in the polynomial form
where a0 = x0, a1=x1 and a2=x2
which ive already shown
so I was thinking that you need to set those for V2 anyway and I can say refer to that problem that this is true and the polynomial form exists for the first 3
oak parrot
so I was thinking that you need to set those for V2 anyway and I can say refer t...
i cannot understand what this means 😖
native carbon
okay so for v2 definition is a3 = .... which means for the a0, a1 and a2 you need to set those yourself since you can't rely on the definition on that yes?
oak parrot
plug in values of n
n=0 yields a0 = c0
n=1 yields a1 = c2+c1+c0
n=2 yields a2 = 4c2 + 2c1 + c0
extrapolate based on this
native carbon
plug in values of n
n=0 yields a0 = c0
n=1 yields a1 = c2+c1+c0
n=2 yields a2 =...
which ive already done 😭 its the same has the x0, x1, x2 thing except rather than saying n=0 it's just x0
oak parrot
okay so for v2 definition is a3 = .... which means for the a0, a1 and a2 you nee...
yes this is where the guess work comes in as I said you need to make an educated guess for your base case
the recursive relationship holds after some point, but not for the first elements in the sequence
native carbon
okay so wait let me just send what i had for showing inclusion the other way and then maybe then my idea will make some sense
oak parrot
the other way is clear already
native carbon
i mean yeah but for induction at least from my understanding you still need to go about it the same way anyway
oak parrot
if you say so
oak parrot
that is correct
native carbon
yes so isnt that just my inductive step
i just need to have the start properly explained
oak parrot
no, this relationship is true by assumption
for all n
you do some algebra to figure out candidate c0 c1 c2, lets call them A,B,C
then your induction assumption is that for some n
an = An^2 + Bn + C
and you must show
an+1 = A(n+1)^2 + B(n+1) + C
native carbon
which for the first 3 would be this
oak parrot
ok, try these
native carbon
plug in values of n
n=0 yields a0 = c0
n=1 yields a1 = c2+c1+c0
n=2 yields a2 =...
okay wait i'm just gonna completely restart my approach and do it as you said here and bring it back to the polynomial form since i think thats just what i need to do idk im starting to confuse myself 😭
harsh zealot
which for the first 3 would be this
digga viel glück
oak parrot
You can also try proving that V_2 is a vector space of dimension 3
also valid
oblique cargo
Yeah but it may be too hard
oak parrot
amounts to roughly the same amount of work
native carbon
You can also try proving that V_2 is a vector space of dimension 3
can you elaborate with what you mean exactly with this
native carbon
yes
oak parrot
are these vector spaces?
oblique cargo
yes
Both being vector spaces (to prove) if you have one inclusion and both having same dimension they would be equal
native carbon
subspaces of this and since they are subspaces they are also vector spaces yes
oak parrot
what is dimension of V1?
native carbon
by the sounds of it 3 but i honestly have no idea why, since this is a sequence
native carbon
oh that's what you mean, yeah in that case
oak parrot
if you were able to show V2 is also dimension 3 then the two would have to be one and the same
harsh zealot
oh that's what you mean, yeah in that case
was willst du studieren?
oblique cargo
You can try proving that (a_n)—->(a_0,a_1,a_2) is an isomorphism from V_2 to R^3
But it’s a possibility, focus on what you started
native carbon
that approach sounds interesting but i think id rather try induction first and then see if i can also show it using a projection
native carbon
was willst du studieren?
ich studiere mathematik :)
oak parrot
that approach sounds interesting but i think id rather try induction first and t...
that approach is actually the standard one
but mutual inclusion is also a valid tactic
oak parrot
fair enough
oak parrot
which for the first 3 would be this
this is your candidate solution
now calculate the induction step
and if it works, you're done
native carbon
hello, had to help my family with something and I'm looking at it again and I still think that what I need to do for the induction step I already did for the other inclusion. I'll write it down all at once and you can tell me where my thinking mistake is or if I'm right
By getting those what you called A,B,C you can say that there exists a n, n+1, n+2 where each of them can be represented in the sequence as the function An^2+Bn+C
so if we then look at an+3 by definition it is 3an+2 - 3an+1 +an
on each of those we can apply the assumption that all of them can be represented as the polynomial function an=An^2+Bn+C though you need to put in the right n for the others of course
After you add it all up you will end with A(n+3)^2+B(n+3)+C which then proves what we were trying to show, that for any given n it can be respresented by that same polynomial function
and that's just what I did for the other inclusion, I really don't understand how else the induction would work rather than just reusing what I already did anyway
only difference is that in one I just inserted the definition of V1 and then it ended up working out but for the other use case you can only do it because of the induction assumption
i mean sure you can put in the A B and C but i don't think it makes a difference whether or not you just leave it as c0 c1 c2, the only important part was that for any given 3 neighbouring an, an+1, an+2 in a sequence there exists a c0, c1 and c2 by which they can be represented with the polynomial function of V1
which is what the induction proof is relying on
abstract tinsel
hello, had to help my family with something and I'm looking at it again and I st...
Correct
abstract tinsel
i mean sure you can put in the A B and C but i don't think it makes a difference...
I think it makes a difference, you need the same A, B and C for the entire sequence
Oh never mind, if you meant just the way you name them then no it doesn't matter
native carbon
yeah it only matters to show that they exist but whether or not you stick to them or just use c0,c1,c2 shouldn't be of relevance
abstract tinsel
So here, a good thing to note is that this reasoning is not basic induction, but strong induction
native carbon
can you elaborate?
abstract tinsel
Usually, in the induction, step, you suppose that a property P(k) is true at a rank k
and try to establish that P(k+1) is true
native carbon
oh thats what you mean
abstract tinsel
In strong induction, you suppose that P(1), ..., P(k) are all true
for a certain k
native carbon
yeah that's what I also noticed, when I was gonna put in the quantor for which n it's valid
but since its valid for every 3 neighbouring an's you could use all
but i just went with "there exists" anyway cuz that's how we usually have it
abstract tinsel
but i just went with "there exists" anyway cuz that's how we usually have it
What is P(n) in your case?
you shouldn't have existential quantifiers in P(n)
native carbon
I'm not sure with the english terminology since I am german 😭
abstract tinsel
I am sorry to inform you that my knowledge of German is limited to "mit Karte bitte"
native carbon
LMAO
nah so what i meant is
when you show that it holds true for the smallest n that you can have you afterwards say that "there is a n for which the statment holds true" yk
abstract tinsel
Indeed, but my question is: what is this statement?
native carbon
oh I can send it let me get my ipad
native carbon
which for the first 3 would be this
i already showed that c0,c1 and c2 exist in iii) which is what my first line is refering to
abstract tinsel
So your induction hypothesis is whatever is after "there exists n in N"?
in the second line
native carbon
yeah
abstract tinsel
then it is incorrect
native carbon
oh what's wrong about it?
abstract tinsel
Because you need to show that the constants don't change throughout the sequence
which you effectively do
but it's not the property you wanted to show
So this is just nitpicking about the logic of the proof, just to be clear
native carbon
which for the first 3 would be this
i mean they are the same ones no? since i already showed it here that they apply for all 3
abstract tinsel
My point is, you need to set up the constants before doing your proof by induction
So for instance:
Let a an element of V2. Let us show that it is an element of V1.
abstract tinsel
which for the first 3 would be this
Let c0 = a0, c1 = (4 a1 - a2 - 3 a0)/2, c2 = (a2 - 2 a1 + a0)/2
We will prove by induction that:
for any n, a_n = c_2 n² + c_1 n + c_0
So here the constants are already fixed
And then, you can use the proof by induction that you showed above
native carbon
oh that's what I was refering to with my first line though
not written properly but i thought that's good enough since for the problem before I had to show that for any x0,x1,x2 you can find a sequene in V1 with c0,c1,c2 where a0=x0 etc
abstract tinsel
oh that's what I was refering to with my first line though
Basically the impression that I had from your proof is that the constants could depend on n
abstract tinsel
I can write it down properly if that eases your mind 😭
No need
I was just nitpicking because you seemed to want a logically sound proof
I admit it might be a bit over the top
native carbon
no no I'd rather have it 100% right than leave some room for an interpretation mistake
I appreciate your point
abstract tinsel
But yeah, that's my two cents
native carbon
earlier it was said that you can also show the inclusion with an isomorphism, do you know how exactly that could be done?
i found that approach interesting but we haven't had isomorphisms for vector spaces yet
native carbon
You can try proving that (a_n)—->(a_0,a_1,a_2) is an isomorphism from V_2 to R^...
this
abstract tinsel
this
Sorry I had to check out of the hotel
So you want to show that V1 = V2
But you already know that V1 is a subset of V2
That is, V1 is a subspace of V2
And now, a subspace is equal to the entire vector space if their dimensions are equal
And two vector spaces are of equal dimension if and only if they are isomorphic
So we just have to find the isomorphism that binds the two
Rotor gave you a suggestion of isomorphism, which you just need to verify is one
native carbon
okay so if have that V1 ⊆ V2 and then show that V2 is isomorph to R^3 how does that lead to V1=V2
i dont follow with that part
so to prove it I understand that if your projection is called idk like P then you need to show that P(an+bn) = P(an)+P(bn) and P(λ*an) = λ*P(an) so how its like with rings and field isomorphism (i assume its the same)
and then for the bijection you just need to show that it's injective and surjective which also seems pretty simple for this example at least
got trouble understanding how showing that leads to being able to say that V1=V2
how I could get it is if you show that V1 -> R^3 is isomorph and V2 -> R^3 is isomorph so V1=V2 should automatically apply
oh wait nevermind he said that you need to show it for both but that was just one example for a projection for V2 -> R^3
okay I think I understand
abstract tinsel
how I could get it is if you show that V1 -> R^3 is isomorph and V2 -> R^3 is is...
Yes
By showing that both V1 and V2 are isomorph to R³, you effectively show that they are both of dimension 3
For V1 it is quite obvious why
For V2 I'd argue it's not that obvious
And that's where Rotor's hint comes in
He gave you the map, you just gotta verify it is an isomorphism
native carbon
i mean i feel like for v2 since you need to show that its isomporph with R^3 to use the starting 3 values for the sequence since the definition only starts for n>=3
i think that's kind of the only way you could go about it anyway but yeah doing it that way by showing the dimension of both being isomorph with R^3 makes sense
thanks a bunch for the help and the pointer with my c0,c1,c2 not being explicitly mentioned to be fixed
I appreciate it a lot
abstract tinsel
i mean i feel like for v2 since you need to show that its isomporph with R^3 to ...
Yes, it is quite clear why the map is linear, but the bijectivity is a bit trickier to justify
abstract tinsel
thanks a bunch for the help and the pointer with my c0,c1,c2 not being explicitl...
You are welcome
native carbon
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