Re: Motivation for matrix algebra

> > Abstract coordinate free linear algebra is one
> > motivation to define matrices and their addition
> and
> > multiplication in the way they are defined. The
> following
> > facts give a guideline:
> >
> > 1. Every finitely generated vector space V (over
> some field
> > K) possesses a finite basis and is thus isomorphic
> to the
> > vector space K^n for some n (called the dimension
> of V).
> >

> I learned that finite vector spaces will have a
> ve a finite basis. When
> you refer to "over some field K", do you mean a field
> to be like
> whether real numbers, integers are used for values of
> the matrices?
>

Not exactly. What I mean is this: a vector space V over
a field K is a set of things (called vectors) that one
can add such that the usual rules of addition are valid.
(Of course one can make this precise writing down
the axioms for addition.)
Moreover one can multiply vectors v with elements of
the field K (called scalars) such that the following
rules hold:
(a+b)v=av+bv
(ab)v=a(bv)
a(v+w)=av+aw
where a,b are elements of K, v,w are elements of V.


>
> > 3. The set Hom(V,W) of all linear maps f:V-->W
> forms
> > a vector space itself if one defines addition of
> linear
> > maps pointwise, and multiplication with a scalar as
> well.
> >
> > Using 2 for fixed bases B,C the vector space
> Hom(V,W)
> > is isomorphic to the space M(mxn,K) of mxn matrices
> > with entries in K, where matrix addition is defined
> in
> > the usual way.
> > Indeed this fact can be considered as the reason
> why
> > addition is defined in the way it is.
> >
> > 4. The set End(V)=Hom(V,W) is a ring: addition is
> defined
> > as in 3 and multiplication is the composition of
> linear
> > maps.
> > Using 2 for fixed basis B,C this ring is ismorphic
> to the
> > ring of square matrices M(nxn,K) with entries in K,
> > where the multiplication is defined in the usual
> way.
> > Again this fact can be considered as the reason to
> define
> > matrix multiplication in the way it is defined.
> >

> I'm unable to follow the above argument since
> since I've been taught
> linear algebra from the applied sciences perspective
> and so do not know
> what Hom(V,W), End(V), M(mxn, K), refer to. However,
> I would guess that
> M is the set of all mxn matrices over the field K,
> and Hom(V,W) is the
> set of all homorphisms from V to W (but I've not
> actually been taught
> homorphisims).
>

That's right. Anyway I defined these sets (in words)
in my original post.


>
> >
> > Remark: it is quite common among students to treat
> > matrices as if they were linear maps. But this
> point of
> > view easily leads to a lot of confusion and to
> > unnecessary work. Rather one should think of
> matrices
> > as coordinate descriptions of linear maps.
>
> Coordinate descriptions of linear maps because the
> the matrices depend
> on which basis used?

Yes.

> I'd appreciate a further
> elaboration because I may
> be one of the students you refer to.
>

Here is an example that hopefully shows the point:

Let V be the set of all polynomial of the form
Ax+B, where A,B are the real coefficients, and x is
the variable of the polynomial.
So V consists of all polynomials in one variable of
degree <=1.

V is a vector space (over the reals), because you can
add polynomials of degree <=1 and get a polynomial of
the same type. Also you can multiply with a real number.

Now recall your analysis course and consider
differentiation: the first derivative (Ax+B)' of a
polynomial in V equals the constant polynomial A, right?

Consider the map:

D: V-->V, p-->p'

that maps a polynomial in V to its first derivative.

This is a linear map as you can easily check.
(However, no matrix appearing here.)

What does a matrix description of the map D look like?

Ok, choose some basis of V. The obvious one is (1,x).
Use this basis >>on both sides of the map V-->V<<,
that is B=C in the notation of my original post.

The coordinate vector of p=Ax+B with respect to (1,x)
is the vector (A,B).
So in particular the coordinate vectors of 1 and x
are (0,1) and (1,0) respectively.
The coordinate vector of d(Ax+B)=A is (0,A).

>From these information you can now derive that the
matrix D representing the linear map d with respect to
the basis (1,x) is the 2x2 matrix

0 0
0 1
.

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