I know that the superscript $T$ on a matrix indicates that the matrix is to be "transposed", that is its columns turned into rows; however for some reason I am seeing it on some solution sets to linear systems, and I am unsure as to why. For example, this is an example of something I have come across:
\begin{equation} \mathrm{x} = (1 - \alpha, 1, \alpha)^{T}, \alpha \in \mathbb{R} \end{equation}
Since this is a solution set and not a matrix, what does this "Transpose" symbol indicate/mean?
You can also transpose a vector. This will turn a row vector into a column vector and vice versa: $$ (1-\alpha, \alpha)^T = \begin{pmatrix}1-\alpha \\ \alpha\end{pmatrix} $$
We often transpose vectors like this, so that we can properly multiply them with a matrix: $$ \begin{pmatrix}a & b \\ c & d\end{pmatrix}(1-\alpha, \alpha)^T $$ is well defined, while $$ \begin{pmatrix}a & b \\ c & d\end{pmatrix}(1-\alpha, \alpha) $$ is not since the dimensions don't match.
edit: For fixed $\alpha\in\Bbb R$ $$ (1-\alpha, \alpha)^T $$ is not a set, but a fixed vector! The solution set is then something like $$ \{(1-\alpha, \alpha)^T : \alpha\in\Bbb R\} = \left\{\begin{pmatrix}1-\alpha \\ \alpha\end{pmatrix} : \alpha\in\Bbb R\right\}. $$