Orthonormal bases

An orthonormal basis consists of mutually orthogonal unit vectors spanning the space.

Different ways to understand it

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Gram–Schmidt process
Independent vectors are orthogonalized by subtracting projections onto earlier directions.
Orthogonal complements
Vectors orthogonal to every vector of a subspace form its orthogonal complement.
QR factorization
A matrix factors into an orthonormal-column factor and an upper-triangular factor under suitable dimensions.
Basis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.
Condition numbers
A matrix condition number describes sensitivity of linear-system solutions to perturbations.