# Is a linear transformation just a mathematical description of a straight line?

On Physics Stack Exchange, the question was asked: Are lorentz transformations linear?

The up-votes given to an answer seemed to be in proportion to how mathematically sophisticated it was, with mine right at the bottom with zero votes; stating that a plot of the transformed variable against the independent untransformed variables is a straight line. After all, linear comes from Latin linearis "belonging to a line,"

Is a linear transformation just a mathematical description of a straight line?

• Instead of saying just straight line, perhaps you should say that the graph of the function is a line, plane or hyperplane. This would then be correct. – Cameron Williams Aug 31 '14 at 17:21
• @CameronWilliams ah, I can see you're right; so I should have said that a linear transformation maps the independent untransformed variables lying on a straight line; to the transformed variable also lying on a straight line. – user10389 Aug 31 '14 at 17:37
• What line is described by the identity $\mathbb{R}^2 \to \mathbb{R}^2$? – Najib Idrissi Oct 31 '14 at 13:26

Excuse me for my bad formalism, I'm no mathematician. Linear is a transformation $T$ from a vector space $X$ to $Y$ when $T(x_1+x_2)=T(x_1)+T(x_2)$ and $T(ax)=a\ T(x)$ if $a$ is a constant, and $x_1$, $x_2$, $x$ are vectors in a vector space $X$. Once again: this is NOT a correct formalization, but purely to grasp the concept of what "linear" in this case means.