Fundamental theorem of linear algebra

The fundamental theorem of linear algebra describes how a matrix’s four fundamental subspaces—the row space, column space, null space, and left null space—relate through orthogonality and dimension.

Row space: The row space of a matrix is the span of its row vectors. It is the orthogonal complement of the null space.
Linear transformation: A linear transformation preserves vector addition and scalar multiplication. A matrix represents such a map between finite-dimensional vector spaces.
Linear system: A linear system is a collection of linear equations in shared unknowns. The column space determines exactly which right-hand sides make the system solvable.
Fredholm alternative: The Fredholm alternative relates solvability of a linear equation to solutions of its adjoint homogeneous equation. It extends the column-space and left-null-space solvability condition beyond finite matrices.
Column space: The column space of a matrix is the span of its column vectors. It is the orthogonal complement of the left null space.
Matrix transpose: The transpose of a matrix is formed by exchanging its rows and columns. Transposing exchanges the roles of the column space and row space.
Least squares: Least squares selects parameters that minimize the sum of squared residuals. The residual lies in the left null space of the coefficient matrix.
Jordan normal form: Jordan normal form represents a complex square matrix using blocks associated with its eigenvalues. It analyzes eigenstructure, whereas the fundamental theorem concerns four subspaces for any matrix.
Null space: The null space of a matrix consists of vectors that it maps to zero. Its vectors are orthogonal to every row of the matrix.
Dimension: The dimension of a vector space is the number of vectors in any basis for it. The theorem states how the dimensions of the four spaces add up.

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Rouché–Capelli theoremRelated: It connects consistency to the column space containing the system’s constants vector.