1xN

In linear algebra, 1xN denotes a matrix with one row and N columns. It is typically interpreted as a row vector in 1-by-N notation. Its entries are a1, a2, ..., aN, and it is often written as A = [a1 a2 ... aN].

In linear algebra, a 1xN matrix is the transpose of an N×1 column vector. The transpose of A is A^T, which is an N×1 column vector (a1, a2, ..., aN)^T. Conversely, any column vector v ∈ R^{N×1} corresponds to the row vector v^T ∈ R^{1×N}.

The rank of a 1xN matrix is 0 if all entries are zero; otherwise it is 1. Since there is only one row, its row space has dimension at most 1, and the matrix encodes a single linear functional on R^N.

Operations: Multiplying A (1×N) on the right by an N×M matrix B yields a 1×M row vector, i.e., AB. The most common use is the dot product with a column vector x ∈ R^{N×1}, which gives a scalar y = A x = ∑ ai xi. The row vector can also be used to act on multiple vectors, producing a set of results.

Example: A = [1, -2, 3], x = [4, 5, -6]^T. Then y = A x = 1*4 + (-2)*5 + 3*(-6) = -24.

Applications: 1×N vectors appear as feature vectors in statistics and machine learning, as weight vectors in linear models, and as representations of data rows in matrices. They are convenient for expressing linear transformations and dot-product computations.