$\begin{pmatrix}a\\b\\c\end{pmatrix}\begin{pmatrix}x&y&z\end{pmatrix}=\begin{pmatrix}ax&ay&az\\bx&by&bz\\cx&cy&cz\end{pmatrix}$ by regular matrix multiplication. But if
$\begin{pmatrix}a\\b\\c\end{pmatrix}$ is an element of the double dual space I believe this product would be a number:
$\begin{pmatrix}a\\b\\c\end{pmatrix}\begin{pmatrix}x&y&z\end{pmatrix}=(cx+by+az)$
Is one of these correct? If I stumble across it what should I do?
Also since I'm teaching this to myself, I tried my best but if I have any other wrong intuition, notation, formulas, or the question is unclear please offer a suggestion so I can fix it.
Edit: I expanded the second product.
Matrix algebra consists of a set of rules for adding and multiplying rectangular arrays of appropriate sizes. According to these rules your first formula is correct.
Matrix algebra is the proper computational machine when working with vectors $x\in V$, where $V$ is a finite dimensional vector space, linear maps $A:\>V\to W$, and functionals $\phi\in V^*$. Common usage treats vectors as "column vectors", functionals (e.g., gradients) as "row vectors", and matrices of linear maps as "acting on the left". In particular the evaluation of a $\phi\in \bigl({\mathbb R}^3\bigr)^*$ on a vector $x\in{\mathbb R}^3$ appears as $$\bigl[\phi_1 \ \phi_2 \ \phi_3\bigr]\>\left[\matrix{x_1\cr x_2\cr x_3\cr}\right]=\phi_1x_1+\phi_2x_2+\phi_3 x_3\ .$$
But matrix algebra with its restriction to arrays of dimension $\leq2$ is not the proper tool when it comes to multilinear algebra where tensors of higher rank, etc., appear. In the same vein, if you consider a vector $x\in V$ as an element in the double dual space, i.e., as a functional on $V^*$, you can no longer expect that the conventions set up in linear algebra 101 can handle this without ado. In this sense your third formula, interpreted as a matrix product, is false even if you correct the typos on the RHS.