Technically, can a matix be 1-Dimensional?

Question

I have this "matrix" and it is 1-dimensional. That is,

$${\bf B} = \mu_1 + \mu_2 - \lambda_A$$

where $\mu_1$, $\mu_2$, and $\lambda_A$ are scalars.

From a technical perspective in mathematics, am I allowed to call this a "matrix"?

Answer 1

(I'm assuming you're referring to real numbers here.) Any scalar $k\in \mathbf{R}$ can be thought of as a $1\times 1$ matrix. This is because it encodes a linear transformation $k:\mathbf{R}\to \mathbf{R}$ given by $k(x)=kx$. In particular, $k(x+y)=kx+ky=k(x)+k(y)$ and for any scalar $\lambda \in \mathbf{R}$, $k(\lambda x)=k\lambda x=\lambda k x=\lambda k(x)$.

So, in short: yes.

As an afterthought, I suppose this should work in any field $\mathbf{F}$ where we think of $\mathbf{F}$ as a $1-$dimensional vector space over itself.