Do points transform differently from vectors?

Question

In this answer, a point $x$ in the coordinate system $C$ transforms to coordinate system $C'$ as follows:

$x' = \frac{x}{2}$

However, the unit vector $i'$ in the $x'$-direction in $C'$ is $2i$ where $i$ is the unit vector in the x-direction in $C$.

What is the reason for this discrepancy? Are points and vectors transformed differently?

Answer 1

Hint:

$(x_1,x_2, \cdots)*({\bf i}, {\bf j} , \cdots)^T = (x'_1,x'_2, \cdots)*({\bf {i'}}, {\bf{ j'}} , \cdots)^T$

so if $({\bf {i'}}, {\bf{ j'}} , \cdots)^T= {\bf T} ({\bf i}, {\bf j} , \cdots)^T$ then ..