How to Extend and Calculate a Basis for the Whole Space

[boderam]

For example, say you start out with (2,1,0) and (2,0,2). Well the easiest answer here is to think of these two vectors in a plane, so you should take the cross product to get the vector that is not in the plane, and there you have a basis for R^3. But how about when we run into similar problems in R^n, not just when we are given n-1 vectors, but perhaps any m less than n. What would be the systematic method?

 

[Tinyboss]

Do a change of coordinates so that the given n vectors are each zero except for a 1 in the n-th position. It should then be obvious how to extend.

 

[boderam]

How do you do the change of basis so this happens? My memory is vague about this.