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Ann
04/17/2022 02:58:45
ok, so then we want a basis for H, yes
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Ann
04/17/2022 02:58:52
first off
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Ann
04/17/2022 02:59:03
let's think about how many elements we should be going for in our basis
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ibzi
04/17/2022 02:59:06
3
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Ann
04/17/2022 02:59:07
ie what dim(H) is
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Ann
04/17/2022 02:59:13
3? you claim dim(H) = 3?
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Ann
04/17/2022 02:59:17
are you sure?
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ibzi
04/17/2022 02:59:17
wait wait wait
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ibzi
04/17/2022 02:59:54
yes
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Ann
04/17/2022 03:00:31
can you tell me what the dimension of P_2 is?
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ibzi
04/17/2022 03:01:03
3?
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Ann
04/17/2022 03:01:38
is that "i think it's 3 but i'm uncertain" or "i like to put question marks after my answer for no reason"
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ibzi
04/17/2022 03:02:12
i think its 3, but that the fact that you've asked this question the 10th time, i'm unsure whether that's true
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Ann
04/17/2022 03:02:47
i only asked you about the dimension of P_2 now. hardly the tenth time.
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Ann
04/17/2022 03:03:15
but yes, the dimension of P_2 is 3, that is correct. you could have corroborated your answer by presenting a basis for P_2 consisting of three elements, such as {1, x, x^2}
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ibzi
04/17/2022 03:03:15
okay well its 3
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Ann
04/17/2022 03:03:33
now think: can dim(H) be 3? clearly H _isn't_ the whole of P_2
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ibzi
04/17/2022 03:04:30
OH
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ibzi
04/17/2022 03:04:51
okay yeah yeah yeah
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ibzi
04/17/2022 03:05:38
wait wait just a second more
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ibzi
04/17/2022 03:06:46
no it could be three
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Ann
04/17/2022 03:07:04
how can dim(H) be 3 without H being the whole space?
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ibzi
04/17/2022 03:07:27
it could be a part of the space but with dim 3
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Ann
04/17/2022 03:07:39
??
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ibzi
04/17/2022 03:08:22
like
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Ann
04/17/2022 03:08:25
okay so let me get this straight: you think that there exists a space V with a subspace W ⊆ V such that dim(V) = dim(W) yet V ≠ W. do i understand you correctly?
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Ann
04/17/2022 03:08:44
yes or no
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ibzi
04/17/2022 03:08:49
yes
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Ann
04/17/2022 03:09:17
can you give me an example of such a subspace for R^3?
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ibzi
04/17/2022 03:09:26
i'm thinking of it
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ibzi
04/17/2022 03:10:35
can't think of anything, maybe its not possible
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Ann
04/17/2022 03:10:35
||because, to put it simply, you are wrong about this. and coming up with such an example is bound to fail. a _proper_ subspace necessarily has _strictly lower_ dimension than its parent space. and pertinent to your problem, this alone means dim(H) ≤ 2.||
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Ann
04/17/2022 03:10:51
then maybe do not make claims you're not 100% certain of lmao
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ibzi
04/17/2022 03:11:20
oh okay okay
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ibzi
04/17/2022 03:11:24
makes sense
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ibzi
04/17/2022 03:11:47
so H would have dim 2 or lower
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Ann
04/17/2022 03:12:09
yes exactly
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Ann
04/17/2022 03:12:14
so now that we know dim(H) ≤ 2
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Ann
04/17/2022 03:12:18
can you give me some elements of H
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Ann
04/17/2022 03:12:33
preferably ones that don't form linearly dependent sets
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ibzi
04/17/2022 03:13:01
um
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ibzi
04/17/2022 03:14:38
{0, 1}, {1, 0}
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Ann
04/17/2022 03:15:38
these are not elements of H
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ibzi
04/17/2022 03:15:43
oh so like
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ibzi
04/17/2022 03:15:55
{1}, {x}
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Ann
04/17/2022 03:15:59
when i ask for elements of H i ask for elements of H
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Ann
04/17/2022 03:16:06
there is also no need to wrap these in braces
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Ann
04/17/2022 03:16:12
now ask yourself: is 1 ∈ H?
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ibzi
04/17/2022 03:16:19
no
1
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ibzi
04/17/2022 03:16:22
x, x^2
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Ann
04/17/2022 03:16:42
is x ∈ H?
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ibzi
04/17/2022 03:16:50
yes
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Ann
04/17/2022 03:16:53
are you sure?
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Ann
04/17/2022 03:17:00
can you state what H is again
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ibzi
04/17/2022 03:17:18
oh it had that property
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ibzi
04/17/2022 03:17:21
so it should be
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Ann
04/17/2022 03:17:34
neither x nor -2 are elements of H...
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Ann
04/17/2022 03:18:03
H = {p ∈ P_2 | **p(2) = 0**}
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ibzi
04/17/2022 03:18:41
yeah so why shouldn't it be x, -2
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ibzi
04/17/2022 03:19:04
p(2) = 0
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Ann
04/17/2022 03:19:11
is x an element of H?
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ibzi
04/17/2022 03:20:05
oh do u mean it should be t?
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ibzi
04/17/2022 03:20:19
because it is p(t)?
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Ann
04/17/2022 03:20:23
i don't care what letter you use for your variable
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Ann
04/17/2022 03:20:33
i mean ok fine i was using x while the problem was using t
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Ann
04/17/2022 03:20:35
doesnt matter
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Ann
04/17/2022 03:21:04
what remains is that i'm trying to get you to name things that are in H, yet the things you are naming are not in H and you for some reason cannot see that
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Ann
04/17/2022 03:21:29
whether you call it x or t, that polynomial is still not an element of H
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ibzi
04/17/2022 03:21:40
that's why i said it
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Ann
04/17/2022 03:21:42
the value of t at t=2 is 2, not 0. H consists of those polynomials whose value at t=2 is 0.
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ibzi
04/17/2022 03:21:45
because it doesn't matter
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ibzi
04/17/2022 03:21:55
yes that's why i added -2
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Ann
04/17/2022 03:22:04
no you didn't
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Ann
04/17/2022 03:22:13
you just said "t and -2"
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ibzi
04/17/2022 03:22:18
yes
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Ann
04/17/2022 03:22:24
as if those were two different things which are individually elements of H
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Ann
04/17/2022 03:22:26
which they are not
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ibzi
04/17/2022 03:22:36
i meant p(t) = t - 2
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Ann
04/17/2022 03:22:42
_then say p(t) = t-2!_
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ibzi
04/17/2022 03:22:53
thats what i've been trying to say the entire time
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ibzi
04/17/2022 03:22:56
._.
1
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Ann
04/17/2022 03:23:06
yet that pesky comma kept getting in the way
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64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:23:12
but ok, great, t-2 is an element of H.
0
1
null
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null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:23:21
can you name another element of H that is not a scalar multiple of t-2?
0
1
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null
null
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1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:23:52
just 0?
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:23:57
or
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:24:00
p(t) = 0
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:24:17
> that is not a scalar multiple of t-2
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:24:22
oh my b
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:24:26
the zero polynomial is in H, but it can't go in a basis.
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:25:44
something like p(t) = t^2 - 4?
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:25:48
no thats the same thing
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:25:49
uh
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
193,692,307,453,771,780
Syst3ms
04/17/2022 03:25:53
No that works
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:25:57
no, t^2 - 4 is not the same thing as t - 2
0
1
183,668,144,404,037,630
Ann
0413
Ann
false
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
193,692,307,453,771,780
Syst3ms
04/17/2022 03:26:04
t+2 isn't a scalar
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
193,692,307,453,771,780
Syst3ms
04/17/2022 03:26:11
there's a t in it
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
380,989,650,615,140,350
ibzi
04/17/2022 03:26:13
oh okay yeah
1
0
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:26:13
t^2 - 4 _is_ in H, and _is not_ a scalar multiple of t-2.
0
1
null
null
null
null
null
1,923
help-10
64bb8913e7b14d06aa9900fca1da50dc
183,668,144,404,037,630
Ann
04/17/2022 03:26:17
it is in fact exactly what i asked for.
0
1
null
null
null
null
null