Mathematics · University
Orthogonality and least-squares fitting · revision notes
The closest point in a subspace is found by an orthogonal residual.
Key ideas and conditions
Projection
The closest point in a subspace is found by an orthogonal residual.
Normal equations
AᵀAx=Aᵀb characterizes least-squares solutions.
Uniqueness
Full column rank makes AᵀA invertible; without it, fitted values may be unique while coefficients are not.
Computation
QR factorization avoids explicitly forming AᵀA and often offers better numerical stability.
Explain without looking
- For data with Sxx=20 and Sxy=−30, find the least-squares slope with an intercept.
- Why is a small least-squares residual not by itself proof of a causal relationship?
Further reading and review references
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