In this post: why do people say “x dimensional vector" a question regarding vector dimension was framed. The response was more or less:
- vectors have dimension equal to space of which the vector is a member, in this case $\mathbb{R}^n$
- rank better describes what is being asked. Vectors have rank 1, Matrices rank 2, Tensors can be any rank.
Given that, if we assign
$$ \vec{b} = \vec{a}^T $$
does $$ \vec{b} == \vec{a} $$
or, more simply, does
$$ \vec{a} == \vec{a}^T $$
In general, no. In order for an $m\times n$ matrix to equal an $m'\times n'$ matrix, you necessarily need $m=m'$ and $n=n'$. For example, the vector $\begin{bmatrix}1\\2\end{bmatrix}$ is a $2\times 1$ matrix but its transpose, $\begin{bmatrix}1&2\end{bmatrix}$, is a $1\times 2$ matrix. Therefore, this vector does not equal its transpose.