Ax1xn

Ax1xn is a compact notation found in some linear algebra texts to denote either the ordered list of images A x1, A x2, ..., A xn under a linear operator A, or the n-by-n matrix whose j-th column is A xj, where x1, ..., xn are basis vectors of a vector space V (often the standard basis in R^n). The usage is not standardized across mathematics and tends to appear as an informal shorthand in lecture notes and certain books to emphasize how A acts on a chosen basis.

Under the interpretation that Ax1xn represents a matrix, Ax1xn is the matrix of A relative to the basis x1, ..., xn. If x1, ..., xn are the standard basis vectors of R^n, then this matrix coincides with A itself. This perspective highlights the connection between applying A to each basis vector and forming the columns of the matrix representing A in that basis.

Example: Let A be the 2×2 matrix [ [a, b], [c, d] ] and let x1 = e1, x2 = e2 be the standard basis of R^2. Then Ax1x2 equals the pair (A e1, A e2) = ( [a, c]^T, [b, d]^T ), and the matrix having these columns is [ [a, b], [c, d] ], which is A. This illustrates how Ax1xn can encode the action of A on a basis as a matrix.

Relation to standard concepts: the notation is closely tied to the idea of representing a linear map by a matrix via its action on a basis, and to the notion of assembling a matrix from column images A xj. The term is most often found in introductory contexts or in notes where concise shorthand is preferred.

See also: Linear transformation, Matrix representation, Basis, Column vectors.