You miss an important part. V and W have to be vector spaces over the same field (or at least the corresponding fields are isomorphic). M_(2x2) is also tricky because most of the time it is called a ring and as a ring it is not isomorphic to R^4 because R^4 is not a ring.
Well we are not talking about rings here, so the fact that it can be understood as a ring with matrix multiplication as the operation does not matter here?
does an isomorphism have to be a linear transformation? if not, couldn't you use a space-filling curve to define an invertible transformation from R to R^2, etc.?
Yes it does. For something to be an isomorphism it must preserve operations by definition. In algebra there is a a concept of homorphism, where the operations are preserved under a function (so if we have a homorphism f and a + b = c, then f(a) + f(b) = f(c), for example.) In linear algebra, these are just linear maps.
It depends what you mean by cardinality. R^2 has the same cardinality as R. That said, if the dimensions have the same cardinality, then I believe the answer is true
I mean... the surface of a sphere is isomorphic to its volume because PhYsIcS (if you added more and more stuff until the universe became a black hole the size of the universe, everything would be represented on the surface of the black hole, therefore the much much sparser real universe must be encodable on a sphere's surface)
i love this guy smiling explaining everything crystal clear.
I have been stuck at this point for such a long time. Thank you so much for offering this wonderful video!
THANK YOU!! GOD BLESS UUU!
ur videos make me realize how many holes are there in my understanding.. seriously!
Thanks we just were given a lecture on isomorphism, please do videos on change of basis
Already on my playlists
Thank you very much.
You miss an important part. V and W have to be vector spaces over the same field (or at least the corresponding fields are isomorphic).
M_(2x2) is also tricky because most of the time it is called a ring and as a ring it is not isomorphic to R^4 because R^4 is not a ring.
Of course
Well we are not talking about rings here, so the fact that it can be understood as a ring with matrix multiplication as the operation does not matter here?
Is dim(ℂ) = 2?
ℝ² isomorphic to ℂ over ℝ
Yes
Depends what your field is.
Dr Peyam I thought the question specified the field to be the reals. In that case the dimensionality is 2, right?
Yeah
Is there any similar criterion for showing that infinite dimensional spaces are isomorphic? Could this extend to uncountable cases?
Appreciate your work :)
Yes,smile makes understanding easy
The Null(T) is basically the Ker(T)?
Yeah
whoa! Dr Peyam rocking the rolex submariner?!
I wish!!! 😂😂😂
does an isomorphism have to be a linear transformation? if not, couldn't you use a space-filling curve to define an invertible transformation from R to R^2, etc.?
Yes it does. For something to be an isomorphism it must preserve operations by definition. In algebra there is a a concept of homorphism, where the operations are preserved under a function (so if we have a homorphism f and a + b = c, then f(a) + f(b) = f(c), for example.) In linear algebra, these are just linear maps.
@@sugarfrosted2005 thank you!
Does the set of 3-dimensional vectors (x, y, 1) qualify as a vector space? Why not?
The vector (0,0,0) does not belong to that set, therefore it isn't a vector space.
If two vectors fields are isomorphic they have the same cardinality? Is the reciprocal also true?
It depends what you mean by cardinality. R^2 has the same cardinality as R. That said, if the dimensions have the same cardinality, then I believe the answer is true
At 7:38 : you must be carefull with the use of "wtf" , its an educational video!
eliya sne ahh humor! 😂
Want to find
Thanks
You’re welcome, Dr 😄
I mean... the surface of a sphere is isomorphic to its volume because PhYsIcS (if you added more and more stuff until the universe became a black hole the size of the universe, everything would be represented on the surface of the black hole, therefore the much much sparser real universe must be encodable on a sphere's surface)
7:30 WTF ?!
Cashman9111 want to find
It's the famous WTF theorem 🙂
happy pride month heres a video about isomorphism 🏳️🌈
Happy pride month 🏳️🌈🏳️🌈🏳️🌈
@@drpeyam A vid about homomorphism would be more appropriate
Pride month is like a square matrix that is not invertible.