## ChapterVVectors

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 (ChapterÂ VS). Initially we will depart from our study of systems of linear equations, but in SectionÂ LC we will forge a connection between linear combinations and systems of linear equations in TheoremÂ 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.

### TheoremÂ 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 (TheoremÂ VSPM) and then this will be generalized into what is arguably our most important definition, DefinitionÂ VS.

### TheoremÂ SLSLC.

Vector addition and scalar multiplication are the two fundamental operations on vectors, and linear combinations roll them both into one. TheoremÂ SLSLC connects linear combinations with systems of equations. This one we will see often enough that it is worth memorizing.

### TheoremÂ 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 SectionÂ ILT which will generalize this result (TheoremÂ KPI).

### TheoremÂ 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 â€” and you always get an answer. Put this one in your toolkit.

### TheoremÂ BNS.

We will have several theorems (all listed in these â€śAnnotated Acronymsâ€ť 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 â€” we will be using it all through the book.

### TheoremÂ 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.