0. Some useful facts

1. Row reduction doesn't change
   • the row space of a matrix,

   • the linear relations between columns of a matrix,

   • the null space of a matrix.

   (Careful--row reduction can change the column space, and usually does.)

2. If you have several vectors such that each one has 1 in some entry for which the others have 0, then they are linearly independent.

   Example: $(1,2,0,3,0)$, $(0,4,1,2,0)$, $(0,3,0,-2,1)$, or the same thing with row vectors or column vectors.

3. A linear combination of column vectors is the same as the matrix made from the columns times the column vector of coefficients.

   Example:

   $x \left[\begin{array}{r}1\\ 2\\ 1\end{array}\right] + y \left[\begin{array}{... ...end{array}\right] \left[\begin{array}{r}x\\ y\\ z\\ s\\ t\end{array}\right]$

   Advice: When you see a linear combination of column vectors like this, you should always think of the matrix form also.