Row and column spaces

The row space of a matrix is the span of its rows, and its column space is the span of its columns. Both have dimension equal to the matrix rank, though they lie in different ambient coordinate spaces for a rectangular matrix.

The row space of a matrix is the linear span of its rows. It is the column space of the matrix transpose.

The column space of a matrix $A$ is the linear span of its columns. It is the image of a linear map represented by $A$.

If an $m\times n$ matrix has columns $c_1,\dots ,c_n$, then matrix multiplication gives $Ax=x_1{c}_1+\cdots +x_n{c}_n$. Thus its column space consists of exactly the attainable output vectors in ${\mathbb{F}}^m$, and $Ax=b$ has a solution exactly when $b$ belongs to this column space. Its dimension is the matrix rank; the kernel instead describes input vectors in ${\mathbb{F}}^n$ whose contributions cancel.^[ref]

A basis of a matrix's column space. Row reduction identifies which original columns to select: take the original columns at the pivot columns of the reduced matrix. Row operations preserve column dependence relations, but generally change the column vectors and their span. Choosing columns of the reduced matrix instead therefore need not give the original column space.