Metamath Proof Explorer

Table of Contents - 11. BASIC LINEAR ALGEBRA

According to Wikipedia ("Linear algebra", 03-Mar-2019, "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part.

Unless otherwise stated, the rings of scalars need not be commutative (see df-cring), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring).

For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space.

  1. Vectors and free modules
    1. Direct sum of left modules
    2. Free modules
    3. Standard basis (unit vectors)
    4. Independent sets and families
    5. Characterization of free modules
  2. Matrices
    1. The matrix multiplication
    2. Square matrices
    3. The matrix algebra
    4. Matrices of dimension 0 and 1
    5. The subalgebras of diagonal and scalar matrices
    6. Multiplication of a matrix with a "column vector"
    7. Replacement functions for a square matrix
    8. Submatrices
  3. The determinant
    1. Definition and basic properties
    2. Determinants of 2 x 2 -matrices
    3. The matrix adjugate/adjunct
    4. Laplace expansion of determinants (special case)
    5. Inverse matrix
    6. Cramer's rule
  4. Polynomial matrices
    1. Basic properties
    2. Constant polynomial matrices
    3. Collecting coefficients of polynomial matrices
    4. Ring isomorphism between polynomial matrices and polynomials over matrices
  5. The characteristic polynomial
    1. Definition and basic properties
    2. The characteristic factor function G
    3. The Cayley-Hamilton theorem