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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?

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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.

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  • $\begingroup$ Does this mean that, whenever we use a matrix to represent a vector there should always be a specified set of basis vectors mentioned as well? $\endgroup$
    – Amin Parvaresh
    Commented Nov 24, 2019 at 15:43
  • $\begingroup$ @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. $\endgroup$
    – Victor Colomb
    Commented Nov 24, 2019 at 19:44

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