4Row rank

We have defined the column rank as the dimension of the space generated by the columns of the matrix. Similarly we can look at the dimension of the space spanned by the rows of a matrix. This number is called the row rank.

We found two numerical values associated to a matrix: Its row rank and its column rank. Now we want to understand how the row rank and the column rank relate to each other.

The question is the following: How do the rows of the matrix (i.e. the different equations of the linear system) affect the dimension of the image of the matrix (i.e. the “size” of the space of vectors y for which the system $Ax=y$ is solvable)? Put differently, can the dimension of the image be determined directly by the relations between the rows of the matrix? And if so, why?