username4417722
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thanks
Last edited by username4417722; 4 weeks ago
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RDKGames
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#2
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(Original post by CaesarAugustus1)
Attachment 1016170
Your result comes from transforming the invariant line y=mx+c and ensuring the result is satisfies this same eqn.

You want lines which do not pass through the origin, i.e. c is non-zero.

But your result you ended up with has to be satisfies for all x. So if c is non-zero, what must m be?

Put this into the coeff of x ... what must lambda be?
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ghostwalker
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(Original post by CaesarAugustus1)
for part a, I have formulated the equation : 0= -2x(m^2 + m + 0.5Lamda ) -2c ( m+1) , however I'm not sure what to do now
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Question is asking for "no invariant lines through the origin".

So, suppose there is an invariant line through the origin.

To start with c will be zero.

Then what about the multiplier for x?

(It's not totally obvious, but have a think)

Note: There is a slicker method of doing this, and also the method you're using doesn't cater for the line x=0, where m would be infinite, and would thus need to be treated separately.


Edit: Since RDKGames is suggesting something different, I suggest going with his method, and you can come back to this one later if you so wish. I'll leave it spoilered for ref.
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ghostwalker
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(Original post by CaesarAugustus1)
0 = -2x(m^2 + m + 0.5Lamda) -2c ( m+1). Seen as c is non zero, m = -1. Therefore -Lamdax = -, and so lamda = 0. But it I'm not sure if that is right
I'm not sure what RDKGames is intending, so can't comment.
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RDKGames
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(Original post by CaesarAugustus1)
0 = -2x(m^2 + m + 0.5Lamda) -2c ( m+1). Seen as c is non zero, m = -1. Therefore -Lamdax = -, and so lamda = 0. But it I'm not sure if that is right
Seems like I wasn’t as clear as I would’ve liked.

Solve for m from coeff of x.

m is not necessarily -1 since it depends on lambda for x coeff to be 0.

So c MUST be zero for an invariant line ...

Ok, so all invariant lines pass through the origin if they exist ... the only thing we can do to ensure there are none passing through origin is to ensure there are no invariant lines at all.

So ... look at m in terms of lambda and decide how you can ensure there are no invariant lines.
Last edited by RDKGames; 4 weeks ago
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RDKGames
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(Original post by CaesarAugustus1)
Lamda is greater than m^2 + m ?
No why?
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ghostwalker
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(Original post by RDKGames)
So c MUST be zero for an invariant line ...

Ok, so all invariant lines pass through the origin if they exist ...
:holmes: A reflection matrix has an infinite number of invariant lines that don't pass through the origin.
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RDKGames
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(Original post by ghostwalker)
:holmes: A reflection matrix has an infinite number of invariant lines that don't pass through the origin.
OP's matrix isn't a reflection.

To be clear, lambda=0 gives infinitely many invariant lines which do not pass through the origin with gradient -1. One of them does (so I suppose this value of lambda can be neglected). For any other lambda c must be zero.

But I am trying to get OP to think about the case of no invariant lines at all since this is an alternative yielding no lines through origin.
Last edited by RDKGames; 4 weeks ago
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RDKGames
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(Original post by CaesarAugustus1)
I am a little confused by what I am trying to do to find the set of values for lamda.
Make the value of m a complex number.
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