Linear Algebra

Chapter TwoVector SpacesThe first chapter began by introducing Gauss’ Method and finished with afair understanding, keyed on the Linear Combination Lemma, of how it findsthe solution set of a linear system. Gauss’ Method systematically takes linearcombinations of the rows. With that insight, we now move to a general study oflinear combinations.We need a setting. At times in the first chapter we’ve combined vectorsfrom R 2 , at other times vectors from R 3 , and at other times vectors from evenhigher-dimensional spaces. So our first impulse might be to work in R n , leavingn unspecified. This would have the advantage that any of the results would holdfor R 2 and for R 3 and for many other spaces, simultaneously.But if having the results apply to many spaces at once is advantageous thensticking only to R n ’s is overly restrictive. We’d like the results to also apply tocombinations of row vectors, as in the final section of the first chapter. We’veeven seen some spaces that are not just a collection of all of the same-sized columnvectors or row vectors. For instance, we’ve seen an example of a homogeneoussystem’s solution set that is a plane, inside of R 3 . This solution set is a closedsystem 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 someof them are in the set.We want the results about linear combinations to apply anywhere that linearcombinations make sense. We shall call any such set a vector space. Our results,instead of being phrased as “Whenever we have a collection in which we cansensibly take linear combinations . . . ”, will be stated as “In any vector space. . . ”.Such a statement describes at once what happens in many spaces. Tounderstand the advantages of moving from studying a single space at a time tostudying a class of spaces, consider this analogy. Imagine that the governmentmade laws one person at a time: “Leslie Jones can’t jay walk.” That would bea bad idea; statements have the virtue of economy when they apply to manycases at once. Or suppose that they ruled, “Kim Ke must stop when passing anaccident.” Contrast that with, “Any doctor must stop when passing an accident.”More general statements, in some ways, are clearer.