How $a_{13}=0$ in $\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$?

Question

I'm reading Artin's Algebra.

$$\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$$

It says that $a_{ij}$ is the matrix entry such that $i$ is the horizontal coordinate and $j$ is the vertical coordinate. It gives some examples: $a_{11}=2, a_{13}=0,a_{23}=5$. I don't understand why $a_{13}=0$ The first number in the first row is $2$, the first number in the third row does not exist. The only way to make this interpretation valid would be to consider the matrix (it would also work for $a_{23}$):

$$\begin{bmatrix} {2}&{1}\\ {1}&{3}\\ {0}&{5} \end{bmatrix}$$

It kinda makes sense, is it possible that when such a situation occur, we just spin the matrix?

Answer 1

It says that $a_{ij}$ is the matrix entry such that $i$ is the horizontal coordinate and $j$ is the vertical coordinate

It does not.

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Answer 2

$a_{13}$ is the rightmost column in the first row (above $a_{23} = 5$), which is clearly $0$. In all other texts, $i$ describes the row (vertical) and $j$ the column (horizontal), it seems like your text got that wrong.

Answer 3

Think of the entries of a matrix as vectors, every row being a vector.

So, $a_{ij}$ is the $j$-th coordinate of the $i$-th vector.

In this case $a_{13}$ is the 3rd coordinate of the 1st vector.

Answer 4

The definition you gave from your book is ambiguous. Here are some other definitions.

The classic book A Survey of Modern Algebra by Birkhoff and MacLane, Third edition, page 159, says:

whose $i$th row consists of the components $a_{i1}, \ldots, a_{in}$

Some other books just give a standard template for a matrix:

$$ \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ a_{21} & \cdots & a_{2n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \\ \end{bmatrix} $$

Your definition is simply wrong. The first coordinate is the row, the second is the column.