Orthogonal complements

Vectors orthogonal to every vector of a subspace form its orthogonal complement.

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Orthonormal bases
An orthonormal basis consists of mutually orthogonal unit vectors spanning the space.
Positive-definite matrices
A symmetric or Hermitian matrix is positive definite when its quadratic form is positive on nonzero vectors.
QR factorization
A matrix factors into an orthonormal-column factor and an upper-triangular factor under suitable dimensions.
Singular values
Singular values are nonnegative square roots of eigenvalues of the conjugate-transpose product.
Basis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.