Chapter V Vectors
We have worked extensively in the last chapter with matrices, and some with vectors. In this chapter we will develop the properties of vectors, while preparing to study vector spaces (Chapter VS). Initially we will depart from our study of systems of linear equations, but in Section LC we will forge a connection between linear combinations and systems of linear equations in Theorem SLSLC. This connection will allow us to understand systems of linear equations at a higher level, while consequently discussing them less frequently.
Annotated Acronyms V.
Theorem VSPCV.
These are the fundamental rules for working with the addition, and scalar multiplication, of column vectors. We will see something very similar in the next chapter (Theorem VSPM) and then this will be generalized into what is arguably our most important definition, Definition VS.
Theorem SLSLC.
Vector addition and scalar multiplication are the two fundamental operations on vectors, and linear combinations roll them both into one. Theorem SLSLC connects linear combinations with systems of equations. This one we will see often enough that it is worth memorizing.
Theorem PSPHS.
This theorem is interesting in its own right, and sometimes the vaugeness surrounding the choice of \(\vect{z}\) can seem mysterious. But we list it here because we will see an important theorem in Section ILT which will generalize this result (Theorem KPI).
Theorem LIVRN.
If you have a set of column vectors, this is the fastest computational approach to determine if the set is linearly independent. Make the vectors the columns of a matrix, row-reduce, compare \(r\) and \(n\text{.}\) That's it — and you always get an answer. Put this one in your toolkit.
Theorem BNS.
We will have several theorems (all listed in these “Annotated Acronyms” sections) whose conclusions will provide a linearly independent set of vectors whose span equals some set of interest (the null space here). While the notation in this theorem might appear gruesome, in practice it can become very routine to apply. So practice this one — we will be using it all through the book.
Theorem BS.
As promised, another theorem that provides a linearly independent set of vectors whose span equals some set of interest (a span now). You can use this one to clean up any span.