Linear Algebra

Matrices, spaces and eigen-structure. The mathematics of systems and transformations: matrices and determinants, solving linear systems and rank, vector spaces and linear maps, eigenvalues and diagonalisation, and orthogonality with LU, Cholesky, QR and SVD — applied to circuits, structures, vibrations and data.

• Matrices & Determinants — Matrices organise systems of numbers and linear maps. Learn matrix types (square, diagonal, identity, symmetric, triangular, orthogonal), addition and multiplication, transpose and conjugate transpose, determinants by cofactor expansion and row reduction, minors and cofactors, the inverse and adjugate, and properties like det(AB) = det A det B.
• Linear Systems & Rank — Solving Ax = b is the most common computation in engineering. Learn augmented matrices, Gaussian and Gauss–Jordan elimination, row echelon forms, rank and the consistency test (Rouché–Capelli), unique/no/infinitely many solutions, homogeneous systems, and conditioning — with a NumPy example.
• Vector Spaces & Linear Transformations — Abstract vector spaces unify vectors, polynomials, signals and functions. Learn vector spaces and subspaces, span, linear independence, basis and dimension, column space, null space and rank–nullity, linear transformations with their matrices, kernel and range, and change of basis.
• Eigenvalues & Eigenvectors — Eigenvectors are directions a matrix only stretches. Learn the characteristic equation, computing eigenvalues and eigenvectors, algebraic and geometric multiplicity, diagonalisation A = PDP⁻¹, symmetric matrices and orthogonal eigenvectors, powers and exponentials of matrices, and engineering uses in vibration modes, stability, principal stresses and PCA.
• Orthogonality & Matrix Decompositions — Factorising matrices is how computers solve linear algebra. Learn inner products and orthogonality, projections and least squares, Gram–Schmidt and QR, LU and Cholesky factorisations, the singular value decomposition (SVD), low-rank approximation, pseudo-inverse and condition number.