How to Interpret Matrices

Question

It seems to me that a m x n matrix can be interpreted in two ways:

1. a collection of m points, each in n-dimensional space. In this case, a point is a row vector.
2. a collection of n points, each in m-dimensional space. In this case, a point is a column vector.

I see in some places they interpret a matrix as 1 and in some places as 2. This causes the notations to be different and makes a lot of thing confusing, at least for me.

Add to this the interpretation of a matrix as a linear transformation and things become more confusing.

I have a few questions, all on the same theme:

1. What is the standard way to interpret matrices?

2. How to reconcile the notations? For example, If I know some theorem for which I had done the derivation understanding a row to be a point and then in some place it says apply the theorem considering a column to be a point. Is there any trick to understand or write equations that makes the two interpretations equivalent?

3. And finally, are the two interpretations equivalent?

Sorry if this is too trivial but this thing seems really confusing.

Answer 1

I'm sorry there's no simple answer for you: matrices appear in many parts of mathematics. How they are interpreted depends on the context. In each case the author should make clear what's meant. Then it's your job to fit that into your abstract understanding. The more you do that the easier it gets.

Answer 2

You mention three interpretations, columns as points, rows as points, and as a representation of a linear transformation. All three interpretations are related, and each has its merits for different uses. I would say that the 'standard' way to think of matrices, if there is such a thing, is that a matrix is a rectangle of numbers. This is a very boring perspective since why should we care about rectangles of numbers. Well, our interest is precisely because of the different interpretations of what a matrix is, which go beyond the immediate visual fact that it is a rectangle of numbers. You choose the 'standard' interpretation based on what you wish to achieve.

As for relating the interpretations. The operation of transposing a matrix interchanges its rows and columns, so that gives you an immediate way to translate between the first two interpretations. When you think of a matrix as representing a linear transformation in the standard bases by means of multiplication on the left, then the columns of the matrix are the values of the linear transformation on the domain standard basis (in the given order). If, instead, you think of a matrix as a linear transformation via multiplication on the right, then the rows of the matrix correspond to the values of the linear transformation on the basis of the domain.

These translation between the interpretations are all helpful when thinking of results in linear algebra. Of course, matrices rely on a choice of basis, and that already dictates a particular approach. I won't elaborate on coordinate-free techniques, but simply mention that it is another possibility for understanding linear algebra. You may experience the plethora of interpretations to be a hinderance at this point, being confused as to which one is most appropriate, but it is in fact a blessing, as hopefully you'll find to appreciate with time.

Answer 3

My preference for viewing matrices (when I don't have a specific reasons for thinking them otherwise) is to interpret them as linear transformations between spaces with a specified basis, and to view the columns as the images of the basis vectors. I wouldn't say that this is the "standard" way, as mathematics gains power from having multiple views of the same situation, but this certainly has more power than just viewing matrices as n-tuples of m-tuples (or vice versa), as you gain matrix multiplication. If you re only going to view a matrix as a collection of values, then you are either ignoring structure present in your problem or there likely wasn't a compelling reason to arrange the numbers into a rectangle.

That said, you can go back and forth between viewing rows or columns as primary quite easily by taking the transpose of the matrix, and one of the confusing things early on in linear algebra is what the transpose actually means.

Suppose that you have a field $K$ and a finite dimensional vector space $K^n$, with basis $e_1, e_2,\ldots, e_n$. The space of linear maps from $K^n$ to $K$ (which we will denote as $(K^n)^*$)is also a vector space of dimension $n$, and we can construct a basis for the new space out of our basis for the old one by defining $e_i^*:K^n\to K$ as $e_i^*(e_j)=1$ if $i=j$ and $0$ otherwise.

Given a linear map $A:K^n\to K^m$, we can construct a map $A^*:(K^m)^*\to (K^n)^*$ by $((A^*)(\varphi))(e)=\varphi(Ae)$, where $e\in K^n$ and $\varphi \in (K^m)^*$. It is a good exercise to verify that this defines a linear map, as the definition can be confusing at first.

Now, the big claim: If we have a basis of $K^m$ and $K^n$ such that $A$ is represented by the matrix $M$, then in terms of the dual bases, $A^*$ will be represented by the transpose of $M$.

I apologize if this construction looks complicated or contrived or ad hoc, but I promise that it actually is very useful and gives a natural situation that swaps rows and columns.

Answer 4

If you have a family of numbers whose index set is a cartesian product you can present this family as a list of lists (Mathematica's way) or as a rectangular array, called a matrix (Matlab's way). Millions of people all over the world work with Excel sheets every day, but they don't think of points in $672$-dimensional space, nor of linear transformations.

An environment where matrices are preminent is linear algebra. Here we have to organize the computations related to instances of the general concepts in a coherent way. This has lead to particular rules and operations with matrices which are not yet present in Excel sheets. E.g., the allocation of the horizontal and the vertical directions to the two index variables is no longer a matter of taste, etc.