Linear algebra underpins machine learning, quantum mechanics, engineering systems, and computer graphics — making it one of the most applied areas of mathematics. Our specialists deliver fully worked solutions across the full syllabus, from row reduction to abstract vector space proofs.
| Computational | Abstract / Theoretical | Applied |
|---|---|---|
| Gaussian elimination and row reduction | Abstract vector spaces (axioms, proofs) | Least squares and regression |
| Matrix operations (inverse, transpose, determinant) | Subspaces, span, basis, dimension | Principal component analysis |
| Eigenvalue and eigenvector computation | Linear independence and dependence proofs | Markov chains |
| Diagonalisation | Linear transformations and their matrices | Computer graphics transformations |
| LU, QR, SVD decompositions | Null space, column space, row space | Network/graph analysis |
| Inner products and orthogonality | Change of basis | Differential equations via matrices |
| Gram-Schmidt orthogonalisation | Spectral theorem | Quantum mechanics (Dirac notation) |
Gaussian elimination must label every row operation explicitly: R₂ → R₂ − 2R₁. Performing operations without labels makes the working unverifiable and loses method marks. Each augmented matrix must be written after every row operation — not just the final result.
Finding eigenvalues requires solving det(A − λI) = 0. This means expanding the determinant correctly, which students often get wrong for 3×3 matrices. The full characteristic polynomial must be written before factoring or solving — not just the roots.
Proving that a set is a subspace requires verifying exactly three things: non-empty (contains zero vector), closed under addition, closed under scalar multiplication. Proving only two of three loses full marks even if the missing one seems obvious. Structure your proofs explicitly.
For diagonalisation, always verify P⁻¹AP = D before presenting it as your answer. A common error is computing eigenvectors incorrectly (especially when eigenvalues are repeated) and assembling a P that does not diagonalise A. This check takes 30 seconds and catches most errors.
Full matrix computations, eigenvalue problems, vector space proofs, and SVD decompositions — with complete working.
Yes. Abstract linear algebra (vector spaces defined over arbitrary fields, linear maps between abstract spaces, dual spaces, tensor products) is handled by our pure mathematicians. These proofs require rigorous axiom-based arguments rather than coordinate computations — a significantly different skill from computational matrix work.
Yes. Numerical linear algebra (condition number, floating point stability, iterative solvers, Krylov methods) appears in applied mathematics and computational science modules. We cover both the theoretical analysis and the MATLAB/Python implementation of these methods.
Yes. The singular value decomposition is one of the most important results in applied linear algebra — used in data compression, image processing, PCA, and pseudo-inverse computation. We deliver both the full derivation of the SVD and its application to specific problems in your assignment.