Explore orthogonal projections onto subspaces of inner product spaces in this 14-minute linear algebra video. Learn about the projection theorem, which demonstrates how to decompose vectors in finite-dimensional subspaces. Discover methods for calculating orthogonal projections of vectors onto subspaces and determining vector components orthogonal to subspaces. Gain a deeper understanding of these concepts through proofs and practical applications. Access additional resources on orthogonal and orthonormal bases, as well as a comprehensive linear algebra course and exercises for further study.