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I'm familiar with the definition of a vector space $V$ over a field $F$

I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space $W$.

The Wiki article on Vector Spaces says this:

Matrices are a useful notion to encode linear maps.

But this conflicts with what I think a matrix is. Is a matrix a "thing" or "element" that we can write down and multiply by other matricies/vectors? Or is it simply a "representation" of a linear map between spaces? Does it depend on the context?

If it is a "thing" - then how do we define the multiplication of objects from the vector space with objects of a different kind that do not belong to the vector space?

We can write a meaningful equation such as:

$M \cdot \textbf{x} = \textbf{y}$

Where x and y are vectors, and M is a matrix.

I know how to perform the multiplication, but it seems wrong to me that $M$ is of a different "species" (apologies - I'm not familiar enough with abstract algebra to know the correct classification - set/group/etc.) than the thing we are multiplying it with.

Sorry if this is vague, I'm just hoping someone might be able to clear up this slight confusion.

Thanks!

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If you imagine a bijection $\phi$ which associates a matrix as a thing in itself with the linear transformation it represents, then you can think of $\mathbf{Mx}=\mathbf y$ as notation for $\phi(\mathbf M)(\mathbf x)=\mathbf y$. –  Rahul Jul 8 '13 at 17:47
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Incidentally, you cannot write $M\cdot x =y$ where $x,y$ are vectors in arbitrary vector spaces of the correct dimension, absent a specific basis for each vector space. –  Thomas Andrews Jul 8 '13 at 17:51
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3 Answers

up vote 3 down vote accepted

To understand the connection between matrix multiplication and linear maps, it is important to understand that a linear map is uniquely determined by what happens to a basis.

To be more specific, consider a linear map $\phi: V \to W $. For the sake of example, let's imagine that $V$ and $W$ are finite dimensional. Say $\dim_{\ F} (V) = n$ and $\dim_{\ F}(W) = m$. Now choose a basis $B=\{v_1, \ldots, v_n\}$ for $V$. I claim that the linear map $\phi$ is uniquely determined by $\phi(v_1), \ldots, \phi(v_n)$. Now for an arbitrary $v \in V$, write $v = c_1 v_1 + \ldots + c_n v_n$ for uniquely determined coefficients $c_i \in F$. Then $\phi(v) = \phi(c_1 v_1 + \ldots + c_n v_n)=c_1\phi(v_1)+ \ldots + c_n\phi(v_n)$, which follows automatically from the linearity of $\phi$.

If you examine the last equation, you should find that this information can be encoded by the multiplication of an $m\times n$ matrix with an $n \times 1$ vector. The columns of the matrix are $\phi(v_1), \ldots, \phi(v_n)$, these expressed in some basis for $W$, while the vector is the coordinates of $v$ in the basis $B$.

Thus, once we have chosen bases for our finite dimensional spaces, there is a perfect correspondence between these linear maps and $m \times n$ matrices. For each choice of bases for $V$ and $W$, we obtain an isomorphism of vector spaces $L(V, W) \cong M_{m \times n}(F)$, where $L(V, W)$ is the space of linear maps from $V$ to $W$ and $M_{m \times n}(F)$ is the space of $m \times n$ matrices with entries in $F$. In fact, this suggests another way to view matrices; they are vectors with a natural multiplication (as long as the dimensions agree).

In some sense, matrix multiplication in this situation is best understood as a window into the inner workings of a linear map. Matrices can of course be seen as independent objects, but there is an inescapable deep connection between their operations and linear maps.

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I think I understand. Since the space of ALL possible linear maps from V to W is isomorphic with the set of ALL m x n matrices, we can work with matrices and perform arithmetic on them and interpret our results in terms of linear mappings because of the isomorphism. So would it be reasonable then to view the "multiplication" of a vector by a matrix just an expression of the matrix's corresponsing map being applied to the vector? I think I'm splitting hairs at this point but I do feel more comfortable thanks to your post. –  user7534 Jul 8 '13 at 18:10
    
Yes, this is a reasonable interpretation. The action of matrices on a vector space via matrix multiplication is in some sense the same as the action of linear maps on that vector space. The equivalence depends on choosing bases. –  James Staff Jul 8 '13 at 18:18
    
It might also be useful to think about things in this way: Choosing bases for $V$ and $W$ amounts to choosing coordinates for those spaces, and matrix multiplication in this sense is what a linear map looks like when expressed in terms of those coordinates. –  James Staff Jul 8 '13 at 18:25
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Note, a matrix does not represent a linear transformation $V\to W$, unless you have fixed bases for $V,W$.

But you are stuck believing that a matrix must be "one" kind of thing. It can be thought of either as a thing of its own, or it can be thought of as a specific linear transformation from $\mathbb k^n\to\mathbb k^m$, and in particular, how it acts on their usual bases.

Picking one view is merely definitional - the two definitions are completely equivalent. Nothing is really gained by treating one of these ideas as "primary," and it is probably better to get used to dichotomy. Sometimes it is useful to think of matrices in their own rights, sometimes it is useful to think of them as linear transformations.

For example, in matrices, it makes much more obvious sense to investigate eigenvalues that are not in the base field $k$. It is harder, initially, to see what that would mean for a linear transformation $T:V\to V$. (It can certainly be defined, it is just not initially obvious.) The fact that these eigenvalues are the same no matter which basis we choose for $V$, however, is really interesting, and thus clearly is a property of the transformation, not the matrix.

On the other hand, a lot of the row calculations that you first learn for matrices are really about changes of bases in the vector space(s).

Sometimes, operations are really "about" the transformation, and sometimes, the operation is deeply tied to the basis. For example, the "transpose" of a matrix is deeply tied to a basis, but the determinant, trace, eigenvalues, rank, etc. of a matrix are independent of the basis. When we really don't care about the basis, we are often dealing with something that is "primarily" a linear transformation.

I suppose the dichotomy starts to have problems when dealing with infinite-dimensional vector spaces.

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If you consistently look at matrices as linear transformations, there is a way to understand multiplication as a transformation as well so they are not "different species".

You may have seen multiplication connected to the AND operation in logic (distributivity), combinatorics (disjoint possibility), probability and the like. The common understanding for matrices is very similar to these (and for very similar reasons). Transformations multiply when you apply one transformation AND THEN the next. Notice that there is a temporal quality to this application.

So, if you follow a point $p$ (seen as a vector from some basis origin) through this transformation, the point after the first transformation $M_1$ is $M_1 p$ AND THEN, after applying another transformation $M_2$ the point is finally $M_2 M_1 p$.

The reason there is this inherent temporal part to the operation is that matrix multiplication is not commutative, nor is the application of linear transformations. The order of applying the transformations matter.

Part of the difficulty in understanding that matrices are associated with transformations is that you may be thinking that matrices need numbers in them and transformations do not need a basis to be meaningful. However, just as you can meaningfully talk about transformations without a basis by describing their properties, you can just as well speak of matrices through the equations they obey without referencing a specific basis or solution. The space of solutions to the equation is the same as the space of transformation representations for transformations with the property.

For example, you might call transformations that applied twice return the space to it's original form as "reflections". You can look at the space of matrices obeying $R^2 = I$ and study the same thing. Similarly, if you call transformations that applied twice are the same as applied only once as "projections", you can study $P^2 = P$ and determine properties of solutions there.

NOTE: the idea of looking at multiplication as AND THEN is taken quite seriously when mathematicians looks at describing logics of transformation. This is found quite a lot in the study of temporal logics, quantum logics, and other nonstandard logic research. The connection then can be seen as an extension of the existing one for standard logic. Addition has a slightly different understanding than one might naively expect from the OR case, but it too has a natural understanding.

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