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Sets will be denoted by braces, { }, between which we will either enumerate the elements of the set or give a rule for determining whether something is an element of the set or not, as in {x | p(x)} , which is read “the set of all x such that p(x) ”.

The way of mathematical thought is twofold: the mathematician first proceeds inductively from the particular to the general and then deductively from the general to the particular. Moreover, throughout its development, mathematics has shown two aspects - the conceptual and the computational - the symphonic interleaving of which forms one of the major aspects of the subject’s aesthetic.

If n > 1 is an integer and if F is a field, it is natural to define addition on the set Fn componentwise:

In general, a vector space does not carry with it the notion of multiplying two vectors in the space to produce a third vector. However, sometimes such multiplication may be possible.

In this chapter we will see how a restricted group of vectors in a vector space over a field can dictate the structure of the entire space, and we will deduce far-ranging conclusions from this.

Let V and W be vector spaces over a field F . A function α : V → W is a linear transformation ^1 or homomorphism if and only if for all.

Let V be a vector space over a field F . A linear transformation a from V to itself is called an endomorphism of V . We will denote the set of all endomorphisms of V by End(V ).

In this chapter we show how we can study linear transformations between finitely-generated vector spaces by studying matrices^1. Let V and W be finitely-generated vector spaces over a field F , where dim( V ) = n and dim( W ) = k .

We are now going to concentrate on the algebraic structure of sets of the form M _nxn( K ), where n is a positive integer and ( K , •) is an associative unital algebra over a field F .

The classical problem of linear algebra is to find all solutions (if any exist) to a system of linear equations in n unknowns of the form