4 ms·
> a matrix operating on a vector always performs an affine transformation on that vector A simple matrix multiplication only performs a linear transformation.
by Agentlien 3y ago
> a matrix operating on a vector always performs an affine transformation on that vector
A simple matrix multiplication only performs a linear transformation. To get a nonlinear affine transformation you need to augment the matrix and vector with extra dimensions containing carefully selected extra ones and zeros in addition to the translation. The end result is still a linear transformation of the supplied vector, but looks like an affine transformation if you ignore the final dimension.
But yes, I did not pick that up in linear algebra. It wasn't even mentioned there. All the fun stuff was taught in computer graphics.
- jasomill 3y agoSpecifically, https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_10-6.pdf https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_1... For more on the underlying geometry, see, e.g., https://archive.org/details/elementarymathem0000klei/page/70/mode/1up https://archive.org/details/elementarymathem0000klei/page/70... or, for a more modern, abstract formulation, https://archive.org/details/geometry0001berg https://archive.org/details/geometry0001berg