# How to properly represent a matrix function.

Given the function $$f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$$, what is the correct way to represent the matrix function in respect to the standard basis?

With the representation theorem, I would write the matrix in columns as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\\ -1 & h & 3\end{bmatrix}$$ But in my textbook it is written in rows as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & -1\\ 0 & 1 & h\\ 1 & 1 & 3\end{bmatrix}$$

What is the difference between them?

The correct way to represent the function $$f_h$$ in matrix form depends on the convention that you want to use to represent it. Let $$(x,y,z) \in \mathbb{R}^3$$. The matrix which you computed is useful for expressing $$f_h$$ as $$f_h(x,y,z) = \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 1 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix}.$$ We can verify this by expanding the matrix product: \begin{align} \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 2 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix} &= \begin{bmatrix} x \cdot 1 + y \cdot 0 + z \cdot (-1) \\ x \cdot 0 + y \cdot 1 + z \cdot h \\ x \cdot 1 + y \cdot 1 + z \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} x - z & y + hz & x + y + 3z \end{bmatrix}. \end{align} The result is a $$1 \times 3$$ matrix, which we can interpret as a row vector in $$\mathbb{R}^3$$. The matrix written in your textbook is useful for the following representation: $$f_h(x,y,z) = \begin{bmatrix} 1& 0 & -1 \\ 0 & 1& h\\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x-z \\ y + hz \\ x + y + 3z \end{bmatrix}.$$ The result is a $$3 \times 1$$ matrix, which we can regard as a column vector in $$\mathbb{R}^3$$.
The two representations look different in the sense that the former yields a row vector and the latter a column vector, however they are the same in the sense that they can both be regarded as lists of three real numbers, i.e. as $$(x-z,y+hz,x+y+3z) \in \mathbb{R}^3.$$ The accepted answer here sums this idea up pretty well.