Linear transformation or not | Mathematics | Page 1

snow perch Apr 28, 2024, 4:47 PM

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is the given transformation linear or not

desert coyoteBOT Apr 28, 2024, 4:47 PM

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snow perch is the given transformation linear or not

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amber burrow Apr 28, 2024, 4:48 PM

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snow perch is the given transformation linear or not

What’s F(0,0)?

snow perch Apr 28, 2024, 4:49 PM

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(0,-1)

amber burrow Apr 28, 2024, 4:50 PM

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Yep it isn’t (0,0)

snow perch Apr 28, 2024, 4:50 PM

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so its not

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right ?

amber burrow Apr 28, 2024, 4:50 PM

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Yes

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Because if it was F(0,0)=(0,0)

snow perch Apr 28, 2024, 4:50 PM

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cause even the transformation of each vectors in a vector addition is not equals to the linear transformation of the whole vector addition

snow perch Apr 28, 2024, 4:51 PM

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amber burrow Because if it was F(0,0)=(0,0)

okay thanks

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also can u help

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with singularity

proven condorBOT Apr 28, 2024, 4:51 PM

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@snow perch has given 1 rep to @amber burrow

amber burrow Apr 28, 2024, 4:51 PM

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Im not familiar with the term does it mean that the null space is non reduced to (0,0)?

snow perch Apr 28, 2024, 4:53 PM

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that only the null subspace can have a null image

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no other subspaces

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can have a null subspace

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and also F inverse transformation also exist

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and F F^-1 = I

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where I is identity

amber burrow Apr 28, 2024, 4:54 PM

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Well it isn’t even a linear transformation so can you even define it

snow perch Apr 28, 2024, 4:54 PM

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hmmm

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yeah

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its not singular

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then

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thankyou sm

snow perch Apr 28, 2024, 4:55 PM

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snow perch where I is identity

also this all

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i said here

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is for non singular

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not for singular

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singular is opposite

amber burrow Apr 28, 2024, 4:56 PM

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Okay I see

snow perch Apr 28, 2024, 4:56 PM

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a non zero subspace may have a null image and inverse F does not exist --> singular

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F F^-1 is not equal to identity

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this all is for singular

amber burrow Apr 28, 2024, 4:57 PM

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It’s basically just that it has a non zero null space

amber burrow Apr 28, 2024, 4:57 PM

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snow perch F F^-1 is not equal to identity

You can’t define F^-1 if F is not bijective null space not 0 => F non bijective

snow perch Apr 28, 2024, 4:58 PM

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ohhhh

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makes sense

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i see

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thanks sm

amber burrow Apr 28, 2024, 4:58 PM

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snow perch makes sense

You’re welcome

snow perch Apr 28, 2024, 4:58 PM

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also can I continue posting questions here , or do i have to create a new forum for that

amber burrow Apr 28, 2024, 4:59 PM

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Hmmm I don’t really know but I think you can post here

snow perch Apr 28, 2024, 4:59 PM

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oh iseee

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thanks

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wait

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amber burrow Apr 28, 2024, 5:01 PM

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Check if there is stability and if each element has an inverse in S

snow perch Apr 28, 2024, 5:01 PM

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but for group i remmeber

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theres just one operation

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and we check its associative under that , additive inverse, identity element

amber burrow Apr 28, 2024, 5:02 PM

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No here you need to check if it’s a group for x

snow perch Apr 28, 2024, 5:02 PM

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snow perch and we check its associative under that , additive inverse, identity element

so i need to check all these but with x

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right ?

amber burrow Apr 28, 2024, 5:03 PM

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You can check if it’s a subgroup of Gl2(C)

amber burrow Apr 28, 2024, 5:03 PM

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snow perch so i need to check all these but with x

Yep

snow perch Apr 28, 2024, 5:03 PM

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i see htankss

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lemme

amber burrow Apr 28, 2024, 5:04 PM

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Calculate A x B

snow perch Apr 28, 2024, 5:04 PM

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here there are 4 elements, but for associative , we just need 3

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smh

snow perch Apr 28, 2024, 5:04 PM

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amber burrow Calculate A x B

oh

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then ?

amber burrow Apr 28, 2024, 5:04 PM

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Well is it in S?

snow perch Apr 28, 2024, 5:04 PM

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lemme chec

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its not

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its making the B matrix

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but interchange

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0 and iota

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that matrix its making

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okay it follows one of

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em

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it has an identity element whihc is the identitty matrix

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lemme chec other

amber burrow Apr 28, 2024, 5:08 PM

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Here you should get A x B=iC

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Which isn’t in S

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So what can you conclude ?

snow perch Apr 28, 2024, 5:09 PM

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uhmmm

snow perch Apr 28, 2024, 5:10 PM

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amber burrow Here you should get A x B=iC

yeah

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uhmm

amber burrow Apr 28, 2024, 5:10 PM

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What properties does a group have to satisfy ?

snow perch Apr 28, 2024, 5:10 PM

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associatve under operation

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existence of identity element under operation

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existence of operative inverse

amber burrow Apr 28, 2024, 5:11 PM

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Yes but you forgot one

snow perch Apr 28, 2024, 5:11 PM

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commutative under operation ( optional)

amber burrow Apr 28, 2024, 5:11 PM

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Stability

snow perch Apr 28, 2024, 5:11 PM

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ohh

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thats not given in our book

amber burrow Apr 28, 2024, 5:11 PM

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So here is it stable under multiplication ?

snow perch Apr 28, 2024, 5:12 PM

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what does stability means

amber burrow Apr 28, 2024, 5:12 PM

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snow perch

What I meant by stability is « closure »

snow perch Apr 28, 2024, 5:12 PM

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amber burrow What I meant by stability is « closure »

ohh

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closuree

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i get it now

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but wasnt it for vector spaces

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and maybe fields as well

amber burrow Apr 28, 2024, 5:13 PM

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So if A and B are in S A x B must be in S

snow perch Apr 28, 2024, 5:13 PM

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ohh I see

amber burrow Apr 28, 2024, 5:13 PM

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snow perch but wasnt it for vector spaces

A vector space is a group under addition

snow perch Apr 28, 2024, 5:13 PM

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yeahh

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its a albenian group

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or whatever that s called

amber burrow Apr 28, 2024, 5:14 PM

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Yep

amber burrow Apr 28, 2024, 5:14 PM

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snow perch its a albenian group

That’s right it’s commutative

snow perch Apr 28, 2024, 5:14 PM

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they have used closure

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in this proof

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but why didnt they tell

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in

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the properties

amber burrow Apr 28, 2024, 5:15 PM

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Well because closure isn’t a property of the group itself but the inner operation

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So x for S

snow perch Apr 28, 2024, 5:15 PM

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I see

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thankss

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lemme do it

amber burrow Apr 28, 2024, 5:16 PM

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You’re very welcome !

snow perch Apr 28, 2024, 5:24 PM

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okay i was afk

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thanks sm tho

amber burrow Apr 28, 2024, 5:25 PM

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You could’ve also proved that the inverse of B isnt in S if you wanted

snow perch Apr 28, 2024, 5:25 PM

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ohh yesss

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its not

amber burrow Apr 28, 2024, 5:26 PM

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Yes

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Both work

deep gyro Apr 28, 2024, 5:27 PM

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snow perch the properties

Yeah, conventions differ on that. Some people list closure as an "axiom", but it's usually redundent. Typeically a group is defined as something like

A group is a set G together with an operation (⋅) on G such that ...
so to make sure a definition is a group, you need to check that (⋅) really is an operation on G (which closure is a part of)

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so to check if a definition is a group, you have to check that the operation is well-defined and satisfies closure, as well as the group axioms

snow perch Apr 28, 2024, 5:46 PM

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oh i seee thanks

#Linear transformation or not