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What's the formal way to map a Matrix $A \in M(n \times n, K)$ to a row vector $B \in K^{n²}$ where

a) the columns

$col_i(A)\quad, \quad 1 \leq i \leq n$

are arranged one below the other

b) the rows

$row_i(A)\quad, \quad 1 \leq i \leq n$

are transposed and then arranged one below the other

How to define B? What's an/the endomorphism from $M(n \times n, K)$ to $K^{n²}$ and backwards?

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up vote 0 down vote accepted

I'm not sure I understand the question, but I'll make a guess. Let $e_{ij}$ be the matrix with a $1$ in row $i$, column $j$, and zeros elsewhere. Let $e_i$ be the vector with a $1$ in component $i$, zeros elsewhere. Define $B$ by $B(e_{ij})=e_{i+(j-1)n}$, and extend $B$ by linearity. That may be the map you want. If not, please clarify your question.

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Any way will do.

What do you mean by the formal way? Any ordering of the $n^2$ numbers in a row is a linear transformation form matrices to vectors.

Since multiplication is not naturally defined on $\mathbb{K}^{n^2}$, the algebraic properties of the matrix group cannot be preserved, so there is no "natural" way of choosing the order of the elements of your vector.

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I'm looking for a way to write $B=...a?...$. I think multiplication is defined on $\mathbb{K}$? – meinzlein Jan 17 '12 at 21:51
It is, but not on $\mathbb{K^{n^2}}$ – yohBS Jan 17 '12 at 22:14

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