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For simplicity I will talk about vector spaces only in a finite number of dimensions. Consider a vector space V and a vector v in that space. Representing v as a matrix in some basis results in a one column matrix. Each vector space has a dual space V' which is the space of all entities that take a vector from V and return a scalar. Consider v' an element of V'. v' acting on v is written as v'(v) - a scalar. Representing v' as a matrix in some basis results in a one row matrix and v'(v) is just the matrix product of the two. I will write repr(v') and repr(v) for the representations of v' and v in the basis in question. To say that v' is the dual of v with respect to the scalar product <,> means that v'(v)=<v,v>. Writing this down with matrices would be repr(v')*repr(v)=repr(v).T*H*repr(v) where H is the hermitian matrix corresponding to the scalar product <,>. Indeed often the basis is chosen such that H is the identity matrix, which means that repr(v).T=repr(v') (i.e. transposing the matrix representing v results in the matrix representing the dual of v) Hopefully I did not introduce any mistakes above. On 25 April 2013 14:01, Prasoon Shukla <prasoon92.i...@gmail.com> wrote: > Okay so let me spell out what I have understood from your comment above, > just to be clear. > > >> If you wish you can think of the dual of a vector (a one-form) as a >> one row matrix (and the vector as a one column matrix). > > > So, if I have column vector v = [ a1, a2, ... , an ] ' (using ' for > transpose), then the dual of v is then a row vector. > >> >> If you have a scalar product operation defined there is a canonical >> way to get a dual of a vector. >> >> For the vector v and scalar product <,>, the oneform for the vector is >> v' such that v'(v) = <v, v> > > > So, let's say H is a nxn positive definite Hermitian matrix, and let us > define the inner product < > of vectors a and b as of dimension n: > <a, b> = Y* H X (where Y and X coordinate matrices of vectors a and b in > some ordered basis) > > Then, from your comment I understand that the dual of a vector v is v' = <v, > v> = V* H V (where V is the coordinate matrix). So it's just an inner > product then? > > But, before you mentioned that the dual of a vector is row matrix, but as > far as I understand, the inner product will just be a scalar, not a row > vector. So, where is it that I am not getting you correctly? Also, I am > assuming that there is a standard Hermitian matrix that is used here, like > maybe the identity matrix perhaps? > > Also, I tried searching a bit more on Google and I found a lot of mention of > the dual space, L(V, F) [ V is a vector space over the field F] which is why > I am forced to believe that there might be some relation between these two > (dual of a vector and dual space). Can you elaborate on this? > > Again I'd like to say that I am really busy with academics for next 3 weeks. > So, if my replies seem to reflect thoughtlessness on my part, please excuse > me for that. I just am really badly occupied. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sympy+unsubscr...@googlegroups.com. > To post to this group, send email to sympy@googlegroups.com. > Visit this group at http://groups.google.com/group/sympy?hl=en-US. > For more options, visit https://groups.google.com/groups/opt_out. > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy?hl=en-US. For more options, visit https://groups.google.com/groups/opt_out.
