Studies linear transformations on finite-dimensional vector spaces \(T: \mathbb{R}^n \to \mathbb{R}^m\). Matrix multiplication \((AB)_{ij} = \sum_{k} A_{ik} B_{kj}\). Determinant \(\det(A)\) gives signed hypervolume scaling; \(\det(A) \ne 0 \iff A\) is invertible. Inverse \(A^{-1} = \frac{1}{\det(A)}\text{adj}(A)\). Gaussian elimination with partial pivoting reduces augmented matrix \([A|I]\) to Reduced Row Echelon Form (RREF), determining rank, nullity, and linear independence.