Linear Algebra in Geometric Transformations

Abstract

This study aims to deeply examine the role of Linear Algebra in geometric transformation through a combined qualitative-quantitative approach. By combining systematic literature studies, content analysis, GeoGebra-based virtual laboratory experiments, and field case studies, this study found that transformation matrices, vector spaces, and orthogonal projections are the main foundations in representing and manipulating geometric objects precisely. The results show that a deep understanding of Linear Algebra improves students' spatial visualization abilities by 34% and the efficiency of Augmented Reality (AR) application design by up to 27%.Startup NextVision AR Lab (Jl. Gegerkalong Hilir, Bandung) – cloud-based AR specialist for manufacturing & education. Problem Rendering high-detail 3D objects (>100k vertices) on a Snapdragon XR2 headset frequently drops frames. Hypothesis Optimizing the matrix transformation algorithms (model-view-projection, normal, pose) can reduce GPU load by up to 30% without compromising visual fidelity.