160 CHAPTER 8. RANK OF A MATRIX

In terms of the original notation, these are the reactions

CO+3H2→ H2O+CH4

O2 +2H2→ 2H2OCO2 +4H2→ 2H2O+CH4

Instead of the four you started with, you could consider the simpler list given above. Theidea is that, in terms of what happens chemically, you obtain the same information withthe shorter list of reactions and have gotten rid of the redundancy which was present in theoriginal list. You can probably imagine that if you had a very large list of reactions madeup from some sort of experimental evidence, such a simplification could be a considerableimprovement.

This is motivation for the general notion of a basis for a vector space which is discussedin the next section. The idea of a basis is similar to what was just done, reducing a list ofreactions to a shorter list. With vectors, you have the span of some vectors and you want toget the shortest possible list of vectors which will leave the span unchanged.

8.5 Linear Independence And Bases8.5.1 Linear Independence And DependenceFirst we consider the concept of linear independence. We define what it means for vectorsin Fn to be linearly independent and then give equivalent descriptions. In the followingdefinition, the symbol, (

v1 v2 · · · vk

)denotes the matrix which has the vector v1 as the first column, v2 as the second column andso forth until vk is the kth column.

Definition 8.5.1 Let {v1, · · · ,vk} be vectors in Fn. Then this collection of vectors is saidto be linearly independent if each of the columns of the n× k matrix(

v1 v2 · · · vk

)is a pivot column. Thus the row reduced echelon form for this matrix is(

e1 e2 · · · ek

)and you cannot delete any of these vectors without diminishing the span of the resultinglist.

The question whether any vector in the first k columns in a matrix is a pivot column isindependent of the presence of later columns. Thus each of {v1, · · · ,vk} is a pivot columnin (

v1 v2 · · · vk

)if and only if these vectors are each pivot columns in(

v1 v2 · · · vk w1 · · · wr

)Here is what the linear independence means in terms of linear relationships.