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    Suppose we have a function f:\mathbb{R}^m \rightarrow \mathbb{R}^n . It's derivative at a \in \mathbb{R}^m is said to be a linear transformation from \mathbb{R}^m \rightarrow \mathbb{R}^n and can be represented by an m x n matrix.

    This is something I have in my notes and is everywhere on the internet. What I don't understnad though is - why is it an m x n matrix. Surely it should be an n x m matrix or else it couldn't act on a vector in R^m ?
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    You're right, it would be an nxm matrix. Are you sure your notes say it would be an mxn matrix?

    I mean, sometimes people write \alpha : \mathbb{R}^n \to \mathbb{R}^m, which does have an mxn matrix representation.
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    A m x n matrix has m rows and n columns. So assuming you write vectors in column form, "m x n" is correct. (Or I'm wrong, of course).
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    (Original post by DFranklin)
    A m x n matrix has m rows and n columns. So assuming you write vectors in column form, "m x n" is correct. (Or I'm wrong, of course).
    With matrix multiplication the number or rows of the matrix on the left must match the number of rows of the matrix on the right. In this case, the matrix on the right is just an m x 1 matrix (using column vectors), so we'd need an n x m matrix.

    EDIT: I'm confused, which is odd given that I use matrices and linear maps almost every day these days. Last year I learnt that (Av)_i = \sum a_{ij} v_j, but in my notes for this year I seem to have [ \alpha (v) ]_i = \sum a_{ji} v_j, which I'd have thought is equal to (vA^T)_i.

    EDIT II: I think I'm getting my bases mixed up.
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    (Original post by nuodai)
    You're right, it would be an nxm matrix. Are you sure your notes say it would be an mxn matrix?

    I mean, sometimes people write \alpha : \mathbb{R}^n \to \mathbb{R}^m, which does have an mxn matrix representation.
    You do IB Analysis II? I've got in my notes that "A \in L(\mathbb{R^m},\mathbb{R^n}) or equvivalently, the space of all mxn matrices [/latex].

    Now I thought that maybe I copied it wrong (or lecturer made a mistake) and it should be an n x m matrix, but then i read this website:

    http://www.colorado.edu/engineering/.../IFEM.AppD.pdf

    which does it the other way round. However, I think that they define their derivative so that u do A^T h in which case this is in line with what I think. SO just to be sure, we have the matrix of size (dimension of range) x (dimension of domain)
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    Yeah, got it 100% wrong. Sorry.
 
 
 
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