How to Extend and Calculate a Basis for the Whole Space

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To extend and calculate a basis for the whole space, one can start with given vectors, such as (2,1,0) and (2,0,2), and use the cross product to find a vector outside their plane, forming a basis for R^3. When dealing with fewer than n vectors in R^n, a systematic method involves changing coordinates so that the vectors have zeros in all but the last position, making extension straightforward. This change of basis simplifies the process of identifying additional vectors needed to complete the basis. Understanding how to perform this change of basis is crucial for effectively extending the set of vectors. Mastery of these techniques is essential for tackling more complex problems in linear algebra.
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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?
 
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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.
 
How do you do the change of basis so this happens? My memory is vague about this.
 

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