Polynomial Transformation Conventions Is there a particular reason why systems of linear equations are compiled as rows in a coefficient matrix, whereas polynomial transformations are compiled as columns in a coefficient matrix. Even then, why do both look at pivots in columns to determine linear independence, shouldn't the polynomial transformation standard matrix have its rows examined to determine linear independence?
Examples:
$x_1 + 2x_2 + 5x_3 = 0$
$7x_1 + x_3= 0$
$-8x_1 + 3x_2 + x_3= 0$
This becomes
\begin{bmatrix}1&2&5\\7&0&1\\-8&3&1\end{bmatrix}
If we assumed the three columns are linearly independent, they form a basis for $\mathbb R^3$:
$$\left\{\begin{bmatrix}1\\7\\-8\end{bmatrix},\begin{bmatrix}2\\0\\3\end{bmatrix}, \begin{bmatrix}5\\1\\1\end{bmatrix}\right\}$$
The vectors in the basis correspond to $x_1, x_2, x_3$
However, this set of polynomials:
$1-2t^2-t^3$
$t+2t^3$
$1+t-2t^2$
Becomes:
\begin{bmatrix}1&0&1\\0&1&1\\-2&0&-2\\-1&2&0\end{bmatrix}
The basis for this is:
$$\left\{1, t, t^2, t^3\right\}$$
Yet $1, t, t^2, t^3$ correspond to rows, rather than columns in the case of the linear trans. matrix
 A: In the context of linear algebra there are two possible conventions to transform linear equations into the framework of matrices resp. linear mappings. 
(1) You consider vectors $x\in\mathbb{R}^d$ as columns, then solving the system of equations $0= \sum_{j=1}^d a_{ij}x_j$ (i=1,...,k) amounts to finding a column vector $x$ with $Ax=0$, where $A$ has exactly the shape you described.
(2) You consider vectors $x\in\mathbb{R}^d$ as rows, then above system of equations is equivalent to $xA^T = 0$, where $A^T$ is the transposed of the matrix above. (Note that here the matrix multiplication also makes sense.)
Both conventions are in use, although the first one is more common, as you have noticed. The reason for this is probably the following: Recall that there is an isomorphism between matrices $A$ and the linear mappings $f_A$ they determine by multiplication. Hence with the first convention $Ax = f_A(x)$, thus in both cases the operator comes first.
This should shed some light on the first part of your question. As for your question about polynomials I am not aware of any conventions like that. In which context have you seen this?
