m×1

m×1 denotes a matrix with m rows and 1 column, commonly interpreted as a column vector of length m. It contains entries x1, x2, ..., xm. In standard linear algebra notation it is written as x = [x1; x2; ...; xm] or as x = (x1, x2, ..., xm)^T, where T denotes transpose. The transpose of an m×1 column vector is a 1×m row vector.

An m×1 vector can be multiplied on the left by a matrix A with dimensions n×m, yielding an n×1 vector Ax. It can be multiplied by a scalar, and two m×1 vectors can be added componentwise. The inner product with another m×1 vector y is x^T y, a 1×1 scalar equal to the sum of the products of corresponding components. The Euclidean norm is ||x|| = sqrt(x^T x).

Interpretation and uses: In R^m, an m×1 column vector represents a coordinate or feature vector. It is the standard convention for state vectors in physics and for coefficients in linear systems. In data science, an m×1 vector can store features of a sample. The 1×m transpose is a row vector; together they can form matrices of other shapes.

Relation to other shapes: A row vector is 1×m, obtained as the transpose of an m×1 vector. Matrices multiply with vectors according to the inner dimension m, so an n×m matrix can multiply an m×1 vector to yield an n×1 result.