Dilations as Vector Transformations

Given a point $O$ and a scale factor $k\ne 0$, a dilation or homothety $H$ is a transformation that maps each point $A$ in space to a point $A'$ such that $\vec{OA'}=k\cdot\vec{OA}$.

To express a dilation as a linear transformation and thus as a matrix, one should first translate the center of homothety to the origin, perform a linear transformation, and then translate the center of homothety back to its original location. This is necessary because in a linear transformation, the origin always maps to itself.

Anyway, let us assume that the origin $O=(0,0,...,0)$ is our center of homothety $H$ in $n$-dimensional space.

Suppose that the scale factor is $k$, and that we have a point $A=(a_1,a_2,...,a_n)$.

Let us solve for our point $A'=H(A)$: $$\vec{OA'}=k\cdot\vec{OA}$$ $$\vec{A'}-\vec{O}=k(\vec{A}-\vec{O})$$ $$\vec{A'}=k\vec{A}$$ Clearly, the vector for $A'$ is simply $k$ times the vector for $A$. Thus, the matrix for the transformation should do nothing but multiply the vector by $k$.

But wait! We already know a matrix that does nothing; namely $I$, the identity matrix! Moreover, to multiply the vector with the scalar $k$, we can just multiply the matrix $I$ by the scalar $k$!

Thus, the matrix for a dilation with center at the origin and scale factor $k$ is simply $kI$: $$kI=\left[ \begin{array}{ccc} k & 0 & 0 & ... & 0 \\ 0 & k & 0 & ... & 0 \\ 0 & 0 & k & ... & 0 \\ ... & ... & ... & ... & ...\\ 0 & 0 & 0 & ... & k \end{array} \right]$$