While studying Linear Algebra from Hoffman Kunze I have following two questions :
I.3. Matrices and Elementary Row Operations
One eannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the "unknowns' $x_{1}, \ldots, x_{n},$ since one actually computes only with the coefficients $A_{i j}$ and the sealars $y_{i} .$ We shall now abbreviate the system (1-1) by $$AX=Y$$ where $$A=\begin{bmatrix}A_{11}&\cdots&A_{1n}\\\vdots&&\vdots\\A_{m1}&\cdots&A_{mn}\end{bmatrix}\\X=\begin{bmatrix}x_1\\\vdots \\x_n\end{bmatrix}\text { and }Y=\begin{bmatrix}y_1\\\vdots\\y_m\end{bmatrix}$$ We call $A$ the matrix of coefficients of the system. Strictly speaking, the rectangular array displayed above is not a matrix, but is a representation of a matrix. An $m \times n$ matrix over the field $F$ is a function $A$ from the set of pairs of integers $(i, j), 1 \leq i \leq m, 1 \leq j \leq n,$ into the field $F$. The entries of the matrix $A$ are the scalars $A(i, j)=A_{i j},$ and quite often it is most convenient to describe the matrix by displaying its entries in a rectangular array having $m$ rows and $n$ columns, as above. Thus $X$ (above) is, or defines, an $n \times 1$ matrix and $Y$ is an $m \times 1$ matrix. For the time being, $A X=Y$ is nothing more than a shorthand notation for our system of linear equations. Later, when we have defined a multiplication for matrices, it will mean that $Y$ is the product of $A$ and $X$
Questions->
(1) Why in 2nd paragraph author wrote that the rectangular array displayed above is not a matrix?
(2) In 2nd line of 2nd paragraph author wrote the definition of matrix. How is a matrix a function from set of pair of integers into the field? To me this defination given contradicts the defination given on Wikipedia and I can't understand it.
Definition on Wikipedia: https://en.m.wikipedia.org/wiki/Matrix_(mathematics)
Can kindly anyone explain why I am confused .
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