If we have a vector that can be represented by an $n$-tuple whose elements can have any values like:
$$ \boldsymbol{a} = (a_1, a_2, \cdots, a_n)$$
we can call $\boldsymbol{a}$ is an $n$-dimensional vector, or the dimension of $\boldsymbol{a}$ is $n$.
However, if some elements of a vector are fixed like:
$$ \boldsymbol{b} = (b_1, b_2, \cdots, b_m, 0, \cdots, 0) $$
$\boldsymbol{b}$ is actually $m$-dimensional, isn't it? In this case, how do we refer to the number of coordinates of $\boldsymbol{b}$, which is larger than $m$?
$$ \mathbf b=(b_1,b_2,\cdots b_m, \underbrace{0,0,\cdots,0}_n) $$ is a vector with $m+n$ components. Since $n$ components are null, the vector $\mathbf b$ is an element of a subspace of dimension $m$ of a vector space of dimension $m+n$.