Question
What do we gain or lose, conceptually, if we consider scalar multiplication as a special form of matrix multiplication?
Background
The question bothers me since I have been reading about dilations and scaling of geometrical objects in Paul Lockhart's book "Measurement". Geometrically, dilation is a transformation that stretches an object in one dimension by a certain factor. Analogously, the linear transformation
$$
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & \lambda
\end{pmatrix}
\cdot
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
$$
"stretches" the third component by the factor $\lambda$. Scaling is a geometric transformation that stretches an object in all dimensions by a certain factor. Analogously, the linear transformation
$$
\begin{pmatrix}
\lambda & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda
\end{pmatrix}
\cdot
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
$$
"stretches" all three components by the factor $\lambda$. This, however, can be written more succinctly using scalar multiplication:
$$
\lambda
\cdot
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}.
$$
In fact, every scalar multiplication can be expressed as a multiplication with a special matrix, and it turns out to be a mere shortcut. On the face of it, this observation is not very spectacular; however, it raises interesting philosophical and conceptual questions as to the foundations of linear algebra.
For example, if scalar multiplication is only a nice-to-have shortcut, then isn't it in fact superfluous conceptually? Currently, scalar multiplication is taught as if it was a distinct concept, independent of matrix multiplication. What would change if we got rid of this shortcut? What could alternative axioms of vector spaces and moduls look like? What about linear transformations? What is easier, what is harder$-$not to write down, but conceptually?
I know that this topic is very broad, but I would like to collect opinions, ideas, examples.
Best Answer
I disagree that "scalar multiplication is only a nice-to-have shortcut", or that is "superfluous conceptually". In fact the very definition of a vector space $V$ requires there to be a scalar multiplication.
After that comes the concept of a linear transformation, which again requires the scalar multiplication to be defined. Matrix multiplication is a convenient way to represent these, and that too only in case of finite dimensional vector spaces (or certain infinite dimensional vector spaces).
So defining matrix multiplication and then saying that scalar multiplication is a special case is putting the cart in front of the horse in my opinion.