Why this written $\vec{v} \binom{x}{y}= \binom{2}{3}$ is not true in mathematics?

Question

My question is trivial but i have asked my self many times why many teachers forbid them students to use this mathematical expression for example $$\vec{v} \binom{x}{y}= \binom{2}{3}$$ but it must $$\vec{v} \binom{2}{3}$$ as a reason that is not true , then my question here is :

Question: Why this written $$\vec{v} \binom{x}{y}= \binom{2}{3}$$ is not true in mathemetic with $\vec{v} $ is a vector in $(O,\vec{i},\vec{j})$

Answer 1

I see one major problem: when the space is not $\mathbb R^p$ or $\mathbb C^p$ since the equality has no sense.

A very minor one: because the column represents the coefficients of $v$ in a basis, and that coefficients depend on the basis. But actually it is not a problem since both should be expressed in the same basis.

Answer 2

First, it must be defined what is "vector". There are multiple ways to define it. Algebraically, it is some object from the set $V$ and associated with the field $\mathbb{K}$ (often, $\mathbb{R}$) s.t. for any $u,v\in V$ and $\alpha\in\mathbb{K}$ operations $u+v$ and $\alpha v$ make sense and, furthermore, some axioms hold. One specific example of $V$ is $\mathbb{R}^n$ which is called arithmetic vector space.

Geometrically, we can think of vectors as objects, that in a given coordinate system can be represented as a column of numbers $(x^1,\dots,x^n)^T$. The objects themselves must be independent of coordinate system and, thus, there are certain rules of how these numbers change with the change of the coordinate system.

Algebraic fact is if you have a vector space $V$ with dimension $n$ on the field $\mathbb{R}$, it is isomorphic to $\mathbb{R}^n$, i.e. for all practical purposes you can think of a vector there as a column of numbers. For example, vectors on the plane are isomorphic to $\mathbb{R}^2$. However, remembering the second definition, we must clarify: in a given coordinate system (basis). So, when we write

$$ \boldsymbol{v}=\left(\begin{matrix} x\\y\end{matrix}\right) $$

in reality we mean

$$ \boldsymbol{v} = x\boldsymbol{e}_1 + y\boldsymbol{e}_2 $$

for some fixed non-collinear non-zero vectors $\boldsymbol{e}_i$.