Linear Algebra › Section F · Orthogonality and least squares › Topic F.3
Orthogonal projections
Given a vector and a subspace, which vector in the subspace is closest? The answer is the orthogonal projection, the shadow the vector casts straight down onto the subspace, and it is the engine behind least squares, data fitting and signal approximation.
In this topic
- The closest point in a subspace
- Projection onto a line
- The orthogonal decomposition theorem
- Why the basis must be orthogonal
- Worked example: projecting onto a plane
- The best approximation theorem
- Worked example: distance to a plane
- Projection matrices from orthonormal bases
- Worked example: a projection matrix
- Common mistakes to avoid
- Key takeaways
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