Should one visualize a matrix by its rows, columns, or both depending on the situation? I see both used and it seems arbitrary. It would be nice if only one was used consistently. Shouldn't a graph of a matrix be denoted as being a row or column representation somehow to avoid confusion?
Example where author switches: https://intuitive-math.club/linear-algebra/matrices
[Example I] Given the transformation:
$$
\begin{bmatrix}
1 & 1\\
2 & 0
\end{bmatrix}
+
\begin{bmatrix}
2 & 1\\
1 & 1
\end{bmatrix}
=
\begin{bmatrix}
3 & 2\\
3 & 1
\end{bmatrix}
$$
The author represents the matrix after the transformation visually by its rows, using the following row vectors:
$$
v_1 =
\begin{bmatrix}
3\\
2
\end{bmatrix}
v_2 =
\begin{bmatrix}
3\\
1
\end{bmatrix}
$$
[Example II] Given the transformation:
$$
\begin{bmatrix}
0 & 1\\
-1 & 0
\end{bmatrix}
\begin{bmatrix}
3 & 1\\
1 & 1
\end{bmatrix}
=
\begin{bmatrix}
1 & 1\\
-3 & 1
\end{bmatrix}
$$
The author represents the matrix after the transformation visually by its columns, using the following column vectors:
$$
v_1 =
\begin{bmatrix}
1\\
-3
\end{bmatrix}
v_2 =
\begin{bmatrix}
1\\
-1
\end{bmatrix}
$$
Question:
Why is did they author seemingly arbitrarily switch from a row → column visual representation? What is the intuition behind this – if any?
Best Answer
There's a lot of ways to interpret matrices, some of which involve reading it by rows and some by columns. But in this particular case, it is columns both times: you were misled by the fact that the matrix $$\begin{bmatrix}3 & 1 \\ 1 & 1\end{bmatrix}$$ is symmetric, so its columns are the same as its rows.
Here, the idea is that for any $2 \times 2$ (or more generally $k \times 2$) matrix $A$, we have $$ A \begin{bmatrix}3 & 1 \\ 1 & 1\end{bmatrix} = \begin{bmatrix}A \begin{bmatrix}3 \\ 1\end{bmatrix} & A\begin{bmatrix}1 \\ 1\end{bmatrix} \end{bmatrix}. $$ In other words, each column of the product is equal to $A$ times a column of the second matrix we multiplied.
In the picture you have, the vector $\begin{bmatrix}3 \\ 1\end{bmatrix}$ (in pink) gets sent to $\begin{bmatrix}1 \\ -3\end{bmatrix}$, and the vector $\begin{bmatrix}1 \\1\end{bmatrix}$ (in yellow) gets sent to $\begin{bmatrix}1 \\ -1\end{bmatrix}$, and all of these are columns of the respective $2 \times 2$ matrix.