Why is the term "$1 \times k$ vector" used rather than $k$-length vector? I see well informed people using the term "$1 \times k$ vector" in lectures. My understanding is that a vector is necessarily $1 \times k$, such that the "$1$ times" part is redundant with "vector." Why is this term used rather than just saying "$k$ length vector"?
 A: In general I do not agree that a vector is necessarily $1$ by $k$, for example in the real vector space $C[0,1],$ the vector $t \mapsto \sin t$ does not in any way seem to be $1$ by $k$. (Unless $k$ represents an uncountable infinity.)
However my guess is that in your lectures, only finite-dimensional vector spaces are discussed. If so, then matrices are probably involved. (True?)
If so, then we might want to distinguish between $1$ by $3$ vectors and $3$ by $1$ vectors, depending on whether we place matrices on the right of the vectors (left) respectively.
For the general $k$-tuple space over a field, the vectors are usually considered to be $1$ by $k$, but it can be convenient to use column vectors i.e. $k$ by $1$.
Technically, the (say the real) $4$-tuple space, composed of elements that look like $(4,27,2021,\pi)$, is a different vector space than the space of $4$ by $1$ real matrices, and is a different vector space than the space of $1$ by $4$ real matrices.
But there is no essential difference, and with appropriate adjustments, we can switch between these three points of view.
