This is a linear algebra review spanning orthogonality, projections, least squares, determinants, and eigenvalues/eigenvectors. Orthonormal bases are characterized by the matrix condition Q^T Q = I, projection matrices project vectors onto lines or subspaces (via P = A(A^T A)^{-1}A^T) and underlie least-squares solutions to inconsistent systems Ax = b, and the Gram-Schmidt process converts an independent basis into an orthogonal (then orthonormal) one by successively subtracting projections onto prior vectors. Determinants are built from three defining properties with further consequences, a signed n-factorial expansion formula, and cofactor expansion (which also yields a matrix-inverse formula), while eigenvalues/eigenvectors satisfy Ax = λx and determine matrix powers, invertibility, stability, and periodicity — including the case of complex-conjugate eigenvalues on the unit circle corresponding to periodic (non-decaying, non-growing) dynamics.
This is a linear algebra review spanning orthogonality, projections, least squares, determinants, and eigenvalues/eigenvectors. Orthonormal bases are characterized by the matrix condition Q^T Q = I, …