Throughout, boldface is used for the row and column vectors. The set of all row vectors forms a vector space called row space, matrices and vectors pdf 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.

Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix. 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. In this case the two matrices are different: they are transposes of each other.

MQ telling us that the matrix product transformation MQ can take v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs. QM v for the composed output from v input.