2×3matris

A 2×3matris is a matrix with 2 rows and 3 columns, consisting of six entries. It is typically denoted by A = [ [a11, a12, a13], [a21, a22, a23] ]. The entries are elements of a field, commonly the real numbers, and the matrix is written as A ∈ F^{2×3}.

As a linear transformation, a 2×3matris represents a map from a three-dimensional input space to a two-dimensional output space. If x is a column vector in F^3, then the product y = A x lies in F^2. Equivalently, A can be viewed simply as a 2×3 array of numbers used in computations or data representations.

The rank of a 2×3matris is at most min(2,3) = 2. The rank indicates the dimension of the row space (spanned by the two rows) and the column space (spanned by the three columns in F^2). If rank(A) = 2, the columns span F^2 and the rows are linearly independent; if rank(A) = 1 or 0, the columns lie in a common line or are zero, respectively.

A concrete example is A = [ [1, 2, 3], [4, 5, 6] ]. Its columns are [1,4], [2,5], and [3,6] in F^2, and the matrix has rank 2, since two columns are linearly independent.

In general, a 2×3matris is not invertible because it is not square. Solving Ax = b often involves least squares when b ∈ F^2 and x ∈ F^3. Such matrices arise in applications where two outputs are produced from three inputs, or where data is organized in two measurements across three features.