# what is the definition of vector components?

I've frequently seen vectors from an n-dim vector space be expressed as what seems to be an n-dim column matrix

$$\mathbf{w}=\begin{pmatrix} w_1 \\w_2 \\ . \\ . \\ w_n \end{pmatrix}$$

and $$w_1, w_2, ..., w_n$$ be called the components of the vector. Is my assumption correct to say that the components of a vector are defined as the coordinates of the vector $$\mathbf{w}$$ w.r.t. the standard basis vectors $$\mathbf{e_i}$$; which would in turn imply that the components of a vector do not change regardless of the basis vectors w.r.t. which it is being expressed?

Let $$(e_1,...,e_n)$$ be a basis of your $$n$$-dim $$\mathbb{K}$$-vector space.
If $$w=\displaystyle\sum\limits_{k=1}^nw_ie_i$$ with $$w_1,...,w_n\in\mathbb{K}$$, then the matrix $$\begin{pmatrix}w_1\\ \vdots\\w_n\end{pmatrix}$$ is said to be associated to the vector $$w$$ using the basis $$(e_1,...,e_n)$$
However, if you choose another basis, the componants $$w_1,...,w_n$$ of the vector $$w$$ may change and therefore the associated matrix too.
• @AminParvaresh Indeed. In some particular cases, the basis may be self-evident in which case there's no need to specify it. For example, in $\mathbb{R}^n$, the basis is naturally $(1,0...),(0,1,0...),...,(0,...,0,1)$, unless specified otherwise. Commented Nov 24, 2019 at 19:44