Three dimensional linear transformations in LaTeX?

Question

I found the method for visualizing two dimensional linear transformations here: Linear transformation Coordinates.

Is there a way to extend this to linear transformations in three dimensions? I imagine there is and that the extension is natural, but I am very unfamiliar with Tikz. Any help would be greatly appreciated.

(Editing to make my question more specific) Is there an extension of the following linear transformation of the plane:

\begin{tikzpicture}
\draw [very thin,gray!40,dashed] (-2,-2) grid (2,2);
\pgftransformcm{0.7}{0.3}{0.9}{0.1}{\pgfpoint{0}{0}}
\draw [black!60] (-2,-2) grid (2,2);
\draw [thick,black,<->] (2.5,0) node [above] {$x$} -- (0,0) -- (0,2.5) node [right] {$y$};
\end{tikzpicture}

to three dimensional space? Specifically, I want to be able to feed in a 3 by 3 matrix and see the transformation of the triangle and axes in the following code.

\begin{tikzpicture}
\draw [->] (0,0,0) -- (2,0,0) node [at end, right] {$x$};
\draw [->] (0,0,0) -- (0,2,0) node [at end, left] {$y$};
\draw [->] (0,0,0) -- (0,0,2) node [at end, left] {$z$};
\draw (1,0,0) -- (0,1,0) -- (0,0,1) -- (1,0,0);
\end{tikzpicture}

I imagine there would be problems if the linear transformation took the triangle out of the first octant, but all matrices I am considering will map the triangle to a subset of itself, so hopefully it's not a problem.

If Tikz won't do it, does anyone know of some software out there for visualizing 3D linear transformations?

Answer 1

I talked to my professor and got a sufficient answer to my question, so I'm putting it here in case others are curious. Since I'm just trying to transform a triangle, I can apply the transformation to the vertices of the triangle, and then the transformed triangle defined by the transformed vertices. This can be computed by hand or with a computer and then put directly into Tikz. No need to make LaTeX do the transformation for me.