Computer Animation Application

Linear Algebra (Matrices) · Axiom Academy

Every animated character on screen is a list of points pushed through a transformation matrix — one multiply per frame. Put that engine in your hands. A game sprite is just a few corner points . To spin it, stretch it, or skew it, the engine multiplies every corner by a single 2×2 matrix — and replaying that multiply as the angle ticks is the whole animation. Push the sprite through a matrix Pick a transform, drag the dial, and every corner of the arrow moves at once — that's one matrix M multiplying each point v to give v' = Mv . Same matrix, every vertex. Where do the basis vectors go? A rotation matrix isn't magic — its two columns are just the new homes of e_1=(1,0) and e_2=(0,1) . Sweep the angle and watch the columns become and . Compose two matrices, then run the frames Scale then rotate is one matrix — multiply them once, apply to the sprite. To animate , just sweep : each frame is the same multiply with a slightly bigger angle. Press Play and watch the frames stream by. Three moves, one idea: apply a matrix to move a shape, read its columns to know what it does, compose & repeat to animate. The same 2×2 (or 3×3) multiply runs game sprites , 3D model rigs in Blender and Maya, camera moves in film VFX, and robot-arm joints — every smooth motion on screen is points pushed through a transformation matrix, one frame at a time.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.