Recent content by ivl

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    From a vector space to the projective space

    Ok, thank you!
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    From a vector space to the projective space

    I, the generator of this post, think that it would perhaps fit better in the category "Geometry and Topology". Can anyone suggest how to relocate a post? The relocation would also help the post get some more attention... Best Wishes, Ivl
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    From a vector space to the projective space

    Thanks for your reply HallsofIvy. Let me formulate the question more precisely, as per your request. In what follows, the vector space V is always 4-dimensional. I would like to proceed like this: -Define an equivalence relation: two vectors in V are equivalent if they differ by a...
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    From a vector space to the projective space

    Dear all, I am not very experienced in this field, so, I have a rather simple question :smile: -Consider a linear vector space V of dimension 4. -Prescribe that, if two vectors in V differ by a nonvanishing constant, they belong to the same equivalence class. -Put together all these...
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    Solbing equation A(u)=B(v) for square matrices A and B

    Thanks everybody, glad to see things got clearer in my head. Problem solved. Cheers IVL
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    Solbing equation A(u)=B(v) for square matrices A and B

    Thanks for your reply, DonAntonio. You are right, my question was a bit too vague. But you understood what I meant. Perhaps a more meaningful way to put the question would be: -given two n times n matrices, A and B -let u and v be two n-dimensional vectors A(u)=B(v) For each...
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    Solbing equation A(u)=B(v) for square matrices A and B

    Dear all, this is perhaps a trivial question, so I apologise in advance. Any help is greatly appreciated nonetheless. ==The Equation== The equation under consideration is: A(u)=B(v) where A and B are n times n matrices, while u and v are n-dimensional vectors. ==The Question==...
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    Casting 2x2 matrix with unit determinant in another form

    Dear Morphism, and dear all, thinking more on the above, I have two questions: 1. Doers there exist an AKA decompostion? (where A and K take the same meaning as in the KAK decomposition) 2. I am very interested in the Bruhat decomposition. Can anyone recommend a good reference for...
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    Casting 2x2 matrix with unit determinant in another form

    Veeery nice! Thanks a lot!
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    Casting 2x2 matrix with unit determinant in another form

    Wow! Thanks Morphism, that is precisely what I was looking for. Do you have any good reference on the Iwasawa decomposition? Cheers IVL Message to other users: for the sake of completeness, if you think there are other decompositions worth being mentioned, please add them to the...
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    Casting 2x2 matrix with unit determinant in another form

    Dear all, I have a simple question for you. Any help will be very appreciated. ==Assumptions=== I have a 2x2 matrix, with real entries: |A B| |C D| Such matrix has unit determinant, AD-BC=1. (For those of you in group theory: the above is a representative of SL(2,R))...
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    Tensor product and infinite dimensional vector space

    I found an answer in wikipedia, see: http://en.wikipedia.org/wiki/Outer_product towards the end of the section "Definition (abstract)".
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    Tensor product and infinite dimensional vector space

    Hi all, it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product. For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}. However, does this...
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    Pull-back of vectors at a SINGLE point

    Thanks Quasar897! the reason I was using the notation ((F_{∗})^{-1}u)_{y} is that I was not aware of the inverse function theorem: xxxxxxx Inverse function theorem xxxxxxxxx If F_{*} is invertible at the point x \in M, then F is a local diffeomorphism around x \in M . In other words...
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    Pull-back of vectors at a SINGLE point

    Dear all, xxxxxxPreliminaryxxxxxxx the push-forward of vectors is FIRST defined at a single point, as (F_{*}v)_{F(x)}(f)=v_{x}(f\circ F) where 1. F_{*} is the push-forward associated with the smooth map F:M\rightarrow N 2. v_{x} is a vector at the point x\in M (a member of the...
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