$\begingroup$Am I correct that a matrix has two dimensions, and a vector has one dimension?
What is the number of dimensions of a scalar? zero?
Thanks.
$\endgroup$11 Answer
$\begingroup$Dimension is a concept that doesn't apply that well to scalars. As Ninad Munshi said in their reply, dimension refers to the vector space in which the matrix/vector is embedded. Vector spaces have a "dimension"; vectors have "rank" (which is basically just the number of elements in the vector; e.g. [3,2] has rank 2 while [7,1,10] has rank 3, etc). Scalars don't really have either.
In other words, a scalar is not simply a rank 1 vector or a rank 1 matrix. Scalars are a different ingredient in the logic of linear algebra. They can be taken from a different space than the vector space (called fields). Often we take our scalars from the Real numbers, which is also where the elements of our vectors/matrices come from in many situations; but this is just coincidence. We could decide "in this scenario, we'll use only Integers to build our vectors, but use Real numbers as our scalars" -- and so on.
To summarize: scalars are different creatures from vectors; they come from a different realm. So, the concepts we use to describe vectors don't really apply to scalars.
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