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A simple question: By definition, does an m x n matrix have m rows and n columns, or is it vice versa?

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    @ColonelPanic, that's probably because for a matrix $A$ operating on an $n$ dimensional vector $\mathcal{x}$ (i.e. $A \mathbf{x} = \mathbf{y}$) $\mathbf{y}$ is $m$ dimensional. In other words, it puts the input dimension before the output dimension alphabetically.2015-04-03

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An $m \times n$ matrix has $m$ rows and $n$ columns.

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    Does this notation convention go back to [Sylvester or Cayley](https://en.wikipedia.org/wiki/Matrix_(mathematics)#History)?2018-03-29
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I suggest you always to check the notation on the book which you are using. I found sometimes this notation with different meaning. In advanced books, for example. Even the notation for linear maps as matrices. Sometimes they write $xT$.

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    It is the notation for the image of $x$ by the linear map $T$. Usually we write $T(x)$ or $Tx$.2012-09-06
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Always check and make sure you have the right convention for the occasion. Usually m x n is rows x columns. I like to remember this as being in REVERSE alphabetical order - Rows by Columns, or R first then C. However, in Boyce & DiPrima's book "Elementary Differential Equations and Boundary Value Problems" an m x n matrix has m vertical columns and n horizontal rows.
However, when addressing elements within a matrix, it's the opposite. The element "a sub i,j" references the element in the ith row and jth column. Lesson? Always check to make sure you have the correct convention!