In Row Vectors Martix - Why can we do column operations? and what is the pattern for system of linear equations in Column Matrix?

Question

In Row Vectors Matrix we have system of linear equations and x,y,z are axes in vector space. $$a_1x + b_1y + c_1z = d$$ $$a_2x + b_2y + c_2z= e$$ $$a_3x + b_3y + c_3z = f$$ $$\begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}d\\e\\f\end{bmatrix}$$ How & Why we treat above matrix for column operation because this has row vectors?

Also in Column Matrix what is the pattern of system of linear equations?

$$\begin{bmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{bmatrix}\begin{bmatrix}?\\?\\?\end{bmatrix} = \begin{bmatrix}?\\?\\?\end{bmatrix}$$ What is the correct way of writing in matrix form also how to write system of linear equation for which can be use for column operation?

Answer 1

Using columns the system becomes

$$xC_1+yC_2+zC_3=D$$ where $C_i$ stands for the $i^{th}$ column.

That is $$x\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix} + y\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}+ z\begin{bmatrix}c_1\\c_2\\c_3\end{bmatrix}= \begin{bmatrix}d\\e\\f\end{bmatrix}$$