What happens if I multiply a point by a scaling matrix?

Question

If I multiply a point by a translation matrix the point is translated. If I multiply a point by a rotation matrix the point is rotated. These two transformations are intuitive. But, when I multiply a point by a scaling matrix it is not scaled (it cannot be), rather it is like "transformed"! This is counter-intuitive! Can anyone please explain this to me? And how is scaling done on a complex 3D model with texture applied on it in computer graphics?

Answer 1

A scaling transformation $T$, as you call it has the property that for $v\in\Bbb R^3$, $Tv = \alpha v$, for some $\alpha \in\Bbb R.$ To see how this looks in matrix form, we use the fact that a linear transformation is determined on a basis, say $e_1, e_2, e_3$, the standard basis. $Te_1 = (\alpha, 0, 0)$, $Te_2 = (0,\alpha, 0)$, $Te_3 = (0,0,\alpha)$. Hence, the matrix $A$ of $T$ is: $$ A=\begin{pmatrix} \alpha & 0 & 0 \\ 0 & \alpha & 0 \\ 0 & 0 & \alpha \end{pmatrix}. $$ So, if you take any point $v\in\Bbb R^3$ and compute $Av$, you will find that the result is equivalent to applying the transformation $\alpha\,\cdot\,\colon\Bbb R^3\to\Bbb R^3$ that "scales" a vector $v$ via standard scalar multiplication. In other words, $Av = \alpha \cdot v$.