The “what” and “why” of linear algebra. Historical review (from systems to vectors, matrices and tensors). Application examples. Vectors and linear combinations. Linear independence. Basic problems of linear algebra as direct or inverse problems of linear combinations. Rules for basic operations with registers and vectors. Solving linear equations. Geometric interpretation by lines, by columns, and an introduction to iterative solving. The concept of erasure, representation of erasure with registers, inversion and permutations. LU factorization.Vector spaces and subspaces. The 4 subspaces defined by a register: Zero space and solving a homogeneous linear system. Registry class. Scalar format of register and reduced lines scalar format. Subspace dimension and bases. Register class factorization. Orthogonality of the four subspaces. Transactions between subspaces. Rectangular complement. Projection and orthogonal projection. Least Squares Approximations. Orthogonal and orthonormal registers. Orthogonal bases and the Gram-Schmidt process. Determinants: Properties, permutations and algebraic complements. Ways to calculate determinant. The geometric meaning of the determinant. Cramer’s rule.Eigenvalues and eigenvectors of register. Diagonalization and Registry Diagonalizability. Similarity of registers. Symmetric and Hermitian registers and their properties. Positively certain registers. Spectral register expansion. Eigenvalue factorization (SVD): Eigenvalues, left and right eigenvectors of register, their properties and their use in dimensionality reduction and applications. Pseudo-reverse registry. The basic factorizations of registers. Non-contestable registers and Jordan form. Linear transformations. Linear transformation register. Base change.