Row and column vectors
In linear algebra, a column vector with elements is an matrix consisting of a single column of entries. Similarly, a row vector is a matrix, consisting of a single row of entries. For example, is a column vector and is a row matrix:
The transpose of any row vector is a column vector, and the transpose of any column vector is a row vector:
Taking the transpose twice returns the original vector:.
The set of all row vectors with entries in a given field forms an -dimensional vector space; similarly, the set of all column vectors with entries forms an -dimensional vector space.
The space of row vectors with entries can be regarded as the dual space of the space of column vectors with entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.
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
Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons.
| Row vector | Column vector | |
| Standard matrix notation | ||
| Alternative notation 1 | ||
| Alternative notation 2 |
Operations
involves the action of multiplying each row vector of one matrix by each column vector of another matrix.The dot product of two column vectors, considered as elements of a coordinate space, is equal to the matrix product of the transpose of with,
By the symmetry of the dot product, the dot product of two column vectors is also equal to the matrix product of the transpose of with,
The matrix product of a column and a row vector gives the outer product of two vectors, an example of the more general tensor product. The matrix product of the column vector representation of and the row vector representation of gives the components of their dyadic product,
which is the transpose of the matrix product of the column vector representation of and the row vector representation of,
Matrix transformations
An matrix can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector, the product is another row vector :Another matrix can act on,
Then one can write, so the matrix product transformation maps directly to. Continuing with row vectors, matrix transformations further reconfiguring -space can be applied to the right of previous outputs.
When a column vector is transformed to another column vector under an matrix action, the operation occurs to the left,
leading to the algebraic expression for the composed output from input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.