Computer Graphics Application

Linear Algebra (Matrices) · Axiom Academy

The Math Behind Every Pixel You Move A shape on screen is just a list of vectors — and rotating, scaling, or re-framing it is one matrix multiply, b = M · v , run on every corner. Every game character, logo, and 3D model is stored as points — and each point is a vector . To rotate it, scale it, or look at it from a new angle, the graphics engine multiplies those vectors by a matrix . Here are the same three moves an engine makes thousands of times a second. Spin the arrow with a rotation matrix Drag the angle and watch every corner move. The rotation matrix multiplies each vertex vector v to give its new spot b — the corners stay the same distance from the origin, the whole shape just turns. Stretch, shrink, and mirror with a scaling matrix A diagonal matrix scales each axis on its own. Push a factor negative and that axis flips — a reflection. The determinant tells you the area factor, and a negative determinant means the shape got mirrored. Object coordinates vs world coordinates — a change of basis A model carries its own axes. Its corners are stored in object coordinates, but the screen needs world coordinates. The bridge is a linear combination of the object's basis vectors — exactly the idea from this unit. Drag the object-space coordinates and watch the point land in the world.

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