📅 2024-12-01 — Session: Gram-Schmidt Process and Projection Matrix Calculation
🕒 00:15–01:30
🏷️ Labels: Gram-Schmidt, Projection Matrix, Linear Algebra, Orthonormal Vectors, Vector Spaces
📂 Project: Teaching
⭐ Priority: MEDIUM
Session Goal
The goal of this session was to execute and reflect on the Gram-Schmidt process and the subsequent calculation of projection matrices in linear algebra.
Key Activities
- Gram-Schmidt Process: Applied the Gram-Schmidt process to initial vectors ( \mathbf{v}_1 ) and ( \mathbf{v}_2 ), resulting in normalized orthonormal vectors.
- Projection Matrix Calculation: Constructed projection matrices using the orthonormal vectors ( w_1 ) and ( w_2 ), allowing for orthogonal projection of vectors in ( \mathbb{R}^4 ).
- Resolution of Linear Systems: Discussed the resolution of linear systems using the projection matrix, analyzing conditions for solutions and their geometric interpretations.
Achievements
- Successfully calculated and interpreted the orthonormal vectors and projection matrices.
- Developed a deeper understanding of the application of projection matrices in solving linear systems.
Pending Tasks
- Further exploration of projection matrices in different vector spaces and their applications in more complex systems.