A transformation function, denoted as $\phi$, maps an input vector $\mathbf{k}_i$ to an output vector $\mathbf{k}'_i$. Below are three examples of this transformation:

1. If $\mathbf{k}_1 = [2, 4]$, the output is $\mathbf{k}'_1 = [4, 16]$.
2. If $\mathbf{k}_2 = [-1, 3]$, the output is $\mathbf{k}'_2 = [1, 9]$.
3. If $\mathbf{k}_3 = [5, 0]$, the output is $\mathbf{k}'_3 = [25, 0]$.

Based on these examples, describe the mathematical operation that the function $\phi$ performs on each component of the input vector to produce the output vector.