What does orthogonal projection refer to in linear algebra?

Orthogonal projection in linear algebra specifically refers to the process of projecting a vector onto another vector in such a way that the resulting projection is the closest point in the direction of the second vector. This concept is crucial because it allows for the decomposition of a vector into two components: one that lies in the direction of the vector onto which the original vector is being projected, and another that lies perpendicular (orthogonal) to it.

In more formal terms, if you have a vector ( \mathbf{v} ) that you want to project onto another vector ( \mathbf{u} ) (assuming ( \mathbf{u} ) is not the zero vector), the orthogonal projection can be calculated using the formula:

[ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} ]

This operation results in a new vector that is aligned with ( \mathbf{u} ), which represents the component of ( \mathbf{v} ) in the direction of ( \mathbf{u} ). The significance of orthogonality here