# Linear Algebra/Vector Spaces

We need a setting for this study. At times in the first chapter, we've combined vectors from $\mathbb {R} ^{2}$ , at other times vectors from $\mathbb {R} ^{3}$ , and at other times vectors from even higher-dimensional spaces. Thus, our first impulse might be to work in $\mathbb {R} ^{n}$ , leaving $n$ unspecified. This would have the advantage that any of the results would hold for $\mathbb {R} ^{2}$ and for $\mathbb {R} ^{3}$ and for many other spaces, simultaneously.
But, if having the results apply to many spaces at once is advantageous then sticking only to $\mathbb {R} ^{n}$ 's is overly restrictive. We'd like the results to also apply to combinations of row vectors, as in the final section of the first chapter. We've even seen some spaces that are not just a collection of all of the same-sized column vectors or row vectors. For instance, we've seen a solution set of a homogeneous system that is a plane, inside of $\mathbb {R} ^{3}$ . This solution set is a closed system in the sense that a linear combination of these solutions is also a solution. But it is not just a collection of all of the three-tall column vectors; only some of them are in this solution set.