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Orthogonality and projection

Created over 8 years ago, updated 10 days ago

Definition: Two vectors (\mathbf{u}) and (\mathbf{v}) are orthogonal if their inner product (dot product) is zero:
[\mathbf{u} \perp \mathbf{v} \iff \langle \mathbf{u}, \mathbf{v} \rangle = 0]

In (\mathbb{R}^n) with the standard dot product:
[\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i = 0]

Orthogonal projection: The projection of (\mathbf{v}) onto a nonzero vector (\mathbf{u}) is:
[\text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}]

This gives the closest point on the line spanned by (\mathbf{u}) to (\mathbf{v}). The error (\mathbf{v} - \text{proj}_{\mathbf{u}}(\mathbf{v})) is orthogonal to (\mathbf{u}).

Example: The projection of ((3,4)) onto ((1,0)) is ((3,0)). The error ((0,4)) is orthogonal to ((1,0)).

Orthogonal decomposition: Any vector (\mathbf{v}) can be written as (\mathbf{v} = \mathbf{v}W + \mathbf{v}{W^\perp}) where (\mathbf{v}W) lies in a subspace (W) and (\mathbf{v}{W^\perp}) is orthogonal to every vector in (W).