TITLE: Is zero a scalar?
QUESTION [11 upvotes]: Is zero considered a scalar?
In other words, is $\begin{bmatrix}0\\0\\\end{bmatrix}$ a scalar multiple of $\begin{bmatrix}a\\b\\\end{bmatrix}$ where $a$ and $b$ are real numbers?

REPLY [10 votes]: As the other answers say, $0$ is indeed a scalar, and $0 {\scriptstyle \begin{bmatrix}a \\ b \end{bmatrix}} = {\scriptstyle \begin{bmatrix}0 \\ 0 \end{bmatrix}}$ is indeed a scalar multiple of ${\scriptstyle \begin{bmatrix}a \\ b \end{bmatrix}}$.  More precisely, whatever field or ring of scalars your matrices are defined over, it surely has a zero element, which we conventionally denote by "$0$".
However, it may be worth noting that the symbol $0$ is also often used to denote a zero matrix, i.e. a matrix whose elements are all zero.  Thus, one might e.g. write $AB = 0$, where $A$ and $B$ are matrices, to indicate that the matrix product of $A$ and $B$ contains all zeros.  This does not usually cause any ambiguity, since it's clear from context what type of entity the symbol $0$ represents in each case.  (Similarly, we often use $I$ to denote "the" identity matrix, without explicitly specifying what the dimensions of this particular identity matrix are.)  Another justification for this notation is that, in the matrix ring of $n \times n$ square matrices, the $n \times n$ zero matrix is indeed the zero element of the ring.
Some authors might use e.g. boldface "$\mathbf 0$" or an underlined "$\underline{\underline 0}$" instead of a plain "$0$" to represent a zero matrix, especially if they follow a convention to highlight all matrix symbols in the same manner.  Similarly, a zero vector might (in contexts where one makes a distinction between vectors and row/column matrices) be denoted e.g. by "$\vec 0$" instead of a plain "$0$".  Many authors, however, especially in more advanced texts, will not bother with such conventions and will just expect you to figure out what type of object "$0$" represents from context.