In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, a column vector or column matrix is an m × 1 matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, i.e. a matrix consisting of a single column of m elements.


The transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 of a column vector is a row vector and vice versa.

The set of all column vectors with a given number of elements forms a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 which is the dual space

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 to the set of all row vectors with that number of elements.

Notation

To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

or

For further simplification, some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with comma

Comma

A comma is a type of punctuation mark . The word comes from the Greek komma , which means something cut off or a short clause.Comma may also refer to:* Comma , a type of interval in music theory...

s and column vector elements with semicolon

Semicolon

The semicolon is a punctuation mark with several uses. The Italian printer Aldus Manutius the Elder established the practice of using the semicolon to separate words of opposed meaning and to indicate interdependent statements. "The first printed semicolon was the work of ... Aldus Manutius"...

s (see alternative notation 2 in the table below).

	Row vector	Column vector
Standard matrix notation
Alternative notation 1
Alternative notation 2

Operations

• Matrix multiplication
  Matrix multiplication
  In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
  
   involves the action of multiplying each row vector of one matrix by each column vector of another matrix.

• The dot product
  Dot product
  In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
  
  of two vectors a and b is equivalent to multiplying the row vector representation of a by the column vector representation of b: