Column vector

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements,

Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements

Throughout, boldface is used for the row and column vectors. The transpose (indicated by T) of a row vector is a column vector

and the transpose of a column vector is a row vector

The set of all row vectors forms a vector space called row space, similarly the set of all column vectors forms a vector space called column space. The dimensions of the row and column spaces equals the number of entries in the row or column vector.

The column space can be viewed as the dual space to the row space, since any linear functional on the space of column vectors can be represented uniquely as an inner product with a specific row vector.

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 (see alternative notation 2 in the table below).

Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix.

The dot product of two vectors a and b is equivalent to the matrix product of the row vector representation of a and the column vector representation of b,

which is also equivalent to the matrix product of the row vector representation of b and the column vector representation of a,

The matrix product of a column and a row vector gives the dyadic product of two vectors a and b, an example of the more general tensor product. The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product,

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