Why is the row space of a matrix important?
The same reason the column space is important.
One example is basic multiplication;
Multiplying a matrix with a vector to the right gives linear combinations of the columns, multiplying with a vector to the left gives linear combinations of the rows.
The row space is used in the fundamental theorem of linear algebra:
which relates row space, column space, kernel and cokernel.
Together, of course, with the rank-nullity theorem.
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