Preservation of distance ratio under an injective linear operator.

Question

Suppose $T$ is a linear operator on $\mathbb R^n$ and $kerT=\{0\}$.Suppose $l$ is a line in the same space and $P$ is a finite set of points on the line,then the distance ratio between points of $P$ will be same as distance ratio of points where elements of $P$ would be mapped under $T$.Is the statement always true?

Answer 1

The image of your line under a linear transformation is:

$T(q+t{v}) = Tq + tTv$

where $q$ is some arbitrary point on the original line, $v$ is a vector and $t$ is a real scalar. If the vector $Tv = u$ is nonzero, then we have yet another line (as opposed to merely a point). Your $T$ has trivial kernel, so the image of the line under $T$ is always another line.

And yes, the distance ratios will be preserved, because the transformed points are still collinear and a linear transformation scales any line uniformly.

See: https://gamedev.stackexchange.com/questions/156864/what-does-a-linear-transformation-preserve

Proof that a linear transformation scales any line uniformly:

Consider two points on the same line, $p_i$ and $p_j$, with

$$p_i = q+t_i v$$ $$p_j = q+t_j v$$

The vector from $p_i$ to $p_j$ is $(t_j-t_i)v$. The magnitude of this distance is $\vert{t_j-t_i}\vert \times \vert\vert v \vert\vert$.

After the transformation we have $Tp_j - Tp_i = (t_j-t_i)Tv$. The magnitude of this distance is $\vert{t_j-t_i}\vert \times \vert\vert Tv \vert\vert$.

We only consider the case where $Tv \ne 0$, so we can take the ratio of distances without dividing by zero:

$$\frac{\vert\vert p_j - p_i \vert\vert}{\vert\vert Tp_j - Tp_i \vert\vert} = \frac{\vert\vert v \vert\vert}{\vert\vert Tv \vert\vert}$$

This distance ratio (the inverse of the scale factor) is independent of where the points $p_i$ and $p_j$ are along the line. It is only dependent on how the vector $v$ (the direction of the line) transforms under $T$. Thus the transformation scales the line uniformly.

Notice that it will scale any line uniformly, but different lines can be scaled by different amounts.