You can do rotations and translations "via linear algebra" without homogeneous coordinates (the 4-element tuple representing a 3d point or vector), since adding two vectors is linear algebra.
What this lets you do is composing multiple transforms into one matrix multiplication instead of a sequence of multiplications and additions; that's what dramatically increases performance, on modern computing devices but most especially on ancient ones, where we were fighting for every MUL.
More details: https://gabrielgambetta.com/computer-graphics-from-scratch/1...
You don't need to use homogeneous coordinates to represent a series of linear transformations as a single matrix. That's just basic linear algebra: any series of linear transformations is itself a linear transformation, and any linear transformation can be represented as an equivalent transformation matrix.
But you do need homogenous coordinates if you want to include translations (fixed distance shifts, such as moving the camera) in the set of operations you can represent as linear transformations and thereby gain all the benefits of linear algebra, including the benefit of being able to include them in a series of operations that you can represent with a single transformation matrix.
Rick Szeliaki’s “Computer vision algorithms and applications” has a lot of great motivation for homogeneous coordinates in the first chapter and appendix. For example, in 2D, the cross product between two points in homogeneous coordinates is the line joining them, and the cross product between two lines in homogeneous coordinates is their point of intersection.
its linear algebra but not a linear transformation. repeat: translations are not linear transformations in 3d.