When I build a representative matrix, I don't understand when to "transpose" its components

Question

From my understanding when I build a representative matrix for a linear transformation I take the transformations product as a linear combination of the bases vectors and place it transposed as the column of the matrix. I accidentally thought that a check for linear dependancy in which the vectors were taken as rows constituted a representative matrix - which they didn't.
here are two examples - I didn't understand the approach to one as opposed to the other.
https://ibb.co/tz4TSbx
https://ibb.co/PGXFhCQ
Edit - clarification: I didn't understand why when writing ImT as is done in the comments we take the vectors as columns while in the solution added below now we use them as rows.
https://ibb.co/HtrR8bN

Answer 1

The second example was to check linear dependancy and not to make a rep matrix, so in that case we do take it as rows and not columns, while when we do work on rep. matrices we always take them as columns. I was confused - hope I didn't mislead anybody or caused to much headaches

Answer 2

I don't see the relation between the first and second picture, since the representative matrix of $T$ should be $3\times3$ (it acts on a three-dimensional vector space). But let's see how to derive the representative matrix of $T$ from the first picture you sent:

Let $T$ be a linear transformation. The representative matrix $A$ of $T$ is built as such:

The $i-$th column of $A$ corresponds to the image of the $i-th$ basis vector by $T$.

For example, as in your picture, let $T: a+bx+cx^2 \mapsto (-2a + b -2c) + (-a -2c)x + (-a + b -3c)x^2$

Your vector space is the space of polynomials of degree $\leq 2$, so a basis is $\{1, x, x^2\}$.

We can see that $T(1) = -2 - x - x^2$, which in the above basis can be written $\begin{pmatrix} -2 \\ -1 \\ -1\end{pmatrix}$.

Similarly, $T(x) = 1 + x^2$, which can be written $\begin{pmatrix} 1 \\ 0 \\ 1\end{pmatrix}$. And $T(x^2)$ is $\begin{pmatrix} -2 \\ -2 \\ -3\end{pmatrix}$.

So the representative matrix of $T$ is $\begin{pmatrix} -2 & 1 & -2 \\ -1 & 0 & -2 \\ -1 & 1 & -3\end{pmatrix}$