[LinAlg] Describe Ker(T) using set notation for T:P^1 -> R

  • #1

Homework Statement


2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation.

Homework Equations


##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x##

The Attempt at a Solution


##T: \mathbb{P}^1 \rightarrow \mathbb{R}## is a mapping/transformation from ##\mathbb{P}^1 \text{ to } \mathbb{R}##.
For ##p(x)## in ##\mathbb{P}^1 \rm , T(p(x))## is the image of ##p(x)## under the action of T. For each p(x) in ##\mathbb{P}^1 \rm , \int_0^1 p(x)~dx## is computed as ##T(p(x)) = Ax = \int_0^1 p(x)~dx = 0##

For p(x) to be mapped to the null space then T(p(x)) must be 0 which means that ##\int_0^1 p(x)~dx=0## which is really ##\int_0^1 a_0 + a_1x~dx = 0##.
Then by integrating we get ##a_0x + \frac 1 2 a_1 x^2 ~|_0^1##.
##a_0 + \frac 1 2 a_1 = 0##
##a_0 = -\frac 1 2 a_1##

Then
##Ker T = \{p(x) = a_0 + a_1x ~|~ a_0 = -\frac 1 2 a_1\}##
 

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  • #2
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Homework Statement


2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation.

Homework Equations


##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x##

The Attempt at a Solution


##T: \mathbb{P}^1 \rightarrow \mathbb{R}## is a mapping/transformation from ##\mathbb{P}^1 \text{ to } \mathbb{R}##.
For ##p(x)## in ##\mathbb{P}^1 \rm , T(p(x))## is the image of ##p(x)## under the action of T. For each p(x) in ##\mathbb{P}^1 \rm , \int_0^1 p(x)~dx## is computed as ##T(p(x)) = Ax = \int_0^1 p(x)~dx = 0##

For p(x) to be mapped to the null space then T(p(x)) must be 0 which means that ##\int_0^1 p(x)~dx=0## which is really ##\int_0^1 a_0 + a_1x~dx = 0##.
Then by integrating we get ##a_0x + \frac 1 2 a_1 x^2 ~|_0^1##.
##a_0 + \frac 1 2 a_1 = 0##
##a_0 = -\frac 1 2 a_1##

Then
##Ker T = \{p(x) = a_0 + a_1x ~|~ a_0 = -\frac 1 2 a_1\}##
That's correct.
 
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  • #3
Ray Vickson
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Homework Statement


2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation.

Homework Equations


##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x##

The Attempt at a Solution


##T: \mathbb{P}^1 \rightarrow \mathbb{R}## is a mapping/transformation from ##\mathbb{P}^1 \text{ to } \mathbb{R}##.
For ##p(x)## in ##\mathbb{P}^1 \rm , T(p(x))## is the image of ##p(x)## under the action of T. For each p(x) in ##\mathbb{P}^1 \rm , \int_0^1 p(x)~dx## is computed as ##T(p(x)) = Ax = \int_0^1 p(x)~dx = 0##

For p(x) to be mapped to the null space then T(p(x)) must be 0 which means that ##\int_0^1 p(x)~dx=0## which is really ##\int_0^1 a_0 + a_1x~dx = 0##.
Then by integrating we get ##a_0x + \frac 1 2 a_1 x^2 ~|_0^1##.
##a_0 + \frac 1 2 a_1 = 0##
##a_0 = -\frac 1 2 a_1##

Then
##Ker T = \{p(x) = a_0 + a_1x ~|~ a_0 = -\frac 1 2 a_1\}##
By ##Ker(T)##, do you really mean ##\text{Ker} (T)##? They look very different! You get the second one by typing "\text{Ker}" instead of "Ker", as required by LaTeX when used properly.

Anyway, you could write ##\text{Ker}(T)## as
$$\bigcup_{a \in \mathbf{R}} \{ p \in \mathbf{P}^1 | p(x) = a - 2ax \}$$
 
  • #5
That's correct.
Great! Thanks! I wasn't quite sure about it. It seemed really simple but really complex at the same time.

Is there a standard style for Ker? ##Ker(T)## is how it's been presented to me.
 
  • #6
Ray Vickson
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Now, now, it's pretty clear what he meant. IMO, this is really nit-picking.

I regard is as a contribution to his education. Maybe he did not know about that before, but now he probably does. Certainly it has done him no harm!
 
  • #7
Ray Vickson
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Great! Thanks! I wasn't quite sure about it. It seemed really simple but really complex at the same time.

Is there a standard style for Ker? ##Ker(T)## is how it's been presented to me.

Probably mistakenly; LaTeX manuals are pretty explicit about such matters, but sometimes people who write notes and lectures may be c
Great! Thanks! I wasn't quite sure about it. It seemed really simple but really complex at the same time.

Is there a standard style for Ker? ##Ker(T)## is how it's been presented to me.

In the AMSMath package there is a command "\ker" that typesets it in the "approved" way, but I don't think the version of LaTeX available in this forum has it. I think the AMSMath version gives ##\text{ker} (T)## instead of ##\text{Ker} (T)##.
 
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  • #8
Probably mistakenly; LaTeX manuals are pretty explicit about such matters, but sometimes people who write notes and lectures may be c

In the AMSMath package there is a command "\ker" that typesets it in the "approved" way, but I don't think the version of LaTeX available in this forum has it. I think the AMSMath version gives ##\text{ker} (T)## instead of ##\text{Ker} (T)##.

Alright. I'll try to keep that in mind then.
 
  • #9
fresh_42
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Great! Thanks! I wasn't quite sure about it. It seemed really simple but really complex at the same time.

Is there a standard style for Ker? ##Ker(T)## is how it's been presented to me.
There are standard functions like \lim, \sin, \log which result in ##\lim, \sin, \log## instead of ##lim, sin, log## which I've written without backslash. However, there are many such conventions which don't have a backslash version. Not sure, whether the kernel is among them. Let's test it:

\ker ##\longrightarrow \ker##
\Ker ##\longrightarrow \Ker##
\im ##\longrightarrow \im##

In case of doubt, i.e. if you don't want to try or look it up, there is a version which always works:
\operatorname{works} results in ##\operatorname{works}## instead of simply ##works##.
 
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  • #12
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Ray Vickson said:
By ##Ker(T)##, do you really mean ##\text{Ker} (T)##? They look very different!
I wouldn't go so far as to say they look "very different."
Is there a standard style for Ker? ##Ker(T)## is how it's been presented to me.
I regard is as a contribution to his education. Maybe he did not know about that before, but now he probably does. Certainly it has done him no harm!
What @bornofflame wrote was clear and unambiguous LaTeX, quite an accomplishment for a fairly new member here. A critque on such a nit as ##Ker(T)## versus ##\text{Ker}(T)## (the latter of which requires twice the typing for a very marginal difference) could at least been more diplomatically presented, in my view. By writing "do you really mean ..." there's the suggestion that the one form is incorrect and the other is correct.
 
  • #13
StoneTemplePython
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For what its worth, I can assure you that others have learned some ideas / techniques from Ray's past formatting pointers. Putting the leading slash in front for things like cosine ##cos(x) \to \cos(x)## is the low hanging fruit.

For non-qualifying ones like ##\text{Ker}(T)## what I've started doing is just writing it as ##Ker(T)## and then at the very end, if I remember,

do a find and replace "Ker(" with "\text{Ker}("

in my editor -- Jupyter notebooks in this case

It really is twice as much work if you do it on your own each each time you type in the term Ker, but almost no more work if you do the find and replace at the very end before submitting your post.
 
  • #14
fresh_42
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I agree with @Mark44. The OP used the kernel twice and both cases could easily be recognized as such. We could as well debate whether the kernel could be denoted with a capital K or not, or if an image should be abbreviated by ##\operatorname{im}## or ##\operatorname{img}##, or whether ##21\,°C## room temperature is better than ##20.5°\,C##.

Edit: Good tip with the replacement, but inconvenient if a) my browser doesn't support it and I don't use an extra editor to write here, or b) there are various locations with different operators. E.g. I regular have ##\operatorname{ad}##, ##\operatorname{Ad}##, ##\operatorname{GL}(V)##, ##\operatorname{ker}##, ##\operatorname{rk}##, etc. Ooops, sorry, I am guilty! I write ##GL(V)## and not ##\operatorname{GL}(V)##. Mea culpa.
 
  • #15
StoneTemplePython
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Edit: Good tip with the replacement, but inconvenient if a) my browser doesn't support it and I don't use an extra editor to write here, or b) there are various locations with different operators. E.g. I regular have ##\operatorname{ad}##, ##\operatorname{Ad}##, ##\operatorname{GL}(V)##, ##\operatorname{ker}##, ##\operatorname{rk}##, etc. Ooops, sorry, I am guilty! I write ##GL(V)## and not ##\operatorname{GL}(V)##. Mea culpa.

I believe you can copy paste your writeup into a regular text editor like Word and do a find and replace all from there. I think any decent one can do that, though the one I use in Ubuntu messes up spacing for some reason?

I don't have any great ideas when there are multiple problem ones -- though a simple python script that runs a through a list of the ones commonly used that aren't "backslashable" comes to mind. If I get around to making such a script maybe I'll share it in the "MATLAB, Maple, Mathematica, LaTeX, Etc" folder...
 
  • #16
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I believe you can copy paste your writeup into a regular text editor like Word and do a find and replace all from there. I think any decent one can do that, though the one I use in Ubuntu messes up spacing for some reason?
Sure, but a change of platforms isn't very convenient.
I don't have any great ideas when there are multiple problem ones -- though a simple python script that runs a through a list of the ones commonly used that aren't "backslashable" comes to mind. If I get around to making such a script maybe I'll share it in the "MATLAB, Maple, Mathematica, LaTeX, Etc" folder...
I installed a little tool which allows me to define my own shortcuts. E.g. ##\operatorname{Ker}## is on my keyboard: Ctrl+o , arrow left, Ker, arrow right. The arrows are, because I encoded \operatorname{} with brackets, because the arrows are much faster to type than the brackets. It's enormously helpful, e.g. for expressions like \left. \dfrac{d}{d}\right|_{}
 

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