Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 5 / a / Solution

For an $m\times n$ matrix $A$ over the real numbers, write its rows as $r_1,\dots ,r_m$ and its columns as $c_1,\dots ,c_n$. ThenThus the row space and column space consist of all linear combinations of the corresponding rows and columns; the null space is the kernel of a linear map represented by $A$. The matrix rank counts independent row or column directions, with row rank equal to column rank.