Final Answer:
The orthogonal projection of a vector onto the subspace spanned by other vectors can be found using the formula P = (v · u) / (||u||²) * u, where P is the orthogonal projection vector, v is the vector being projected, and u is a vector spanning the subspace.
Step-by-step explanation:
The orthogonal projection is a fundamental concept in linear algebra, particularly when dealing with subspaces. Given a vector v and a subspace spanned by vectors, represented by u, the orthogonal projection P of v onto the subspace can be calculated using the formula mentioned above.
In the formula, the dot product (v · u) represents the projection of v onto u, and ||u||² is the squared magnitude of u. The division ensures that the resulting projection vector P is scaled appropriately. The projection vector P is the component of v that lies in the subspace spanned by u, and it is orthogonal to the complement of the subspace.
This projection process is valuable in various applications, such as signal processing, computer graphics, and machine learning. It allows for the decomposition of a vector into components that align with a given subspace, providing insights into the structure and relationships within a vector space.