1. The row space of a matrix

Problem. Find a basis for the row space of

$M = \left[\begin{array}{rrrrr}1&2&1&2&3\\ 2&4&3&7&8\\ 1&2&3&8&8\end{array}\right]$.

Method. Row-reduce and take the nonzero rows. They have the same span as the original rows and are also linearly independent.

Note. This is also a way to find a basis for the span of a list of vectors: Make a matrix with those vectors as rows and row reduce. (See also section .)