An ordered set of vectors forms a basis for a vector space if it spans its space using the minimum number of vectors needed to do so.
Definition:
If \(S = \{ \overrightarrow{v_1}, \overrightarrow{v_2} ,\dotsc, \overrightarrow{v_n} \} \)is a set of vectors in a finite-dimensional vector space \( V \), then \( S \) is called a basis for \( V \) if:
(1) \( S \) spans \( V \)
(2) \( S \) is linearly independent
Example:
Consider the two following sets of vectors.
\( A = \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \} \qquad B = \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \} \)
The set \(A \) does not form a basis for \( \mathbb{R^2} \) because although it spans \( \mathbb{R^2} \), it is not linearly independent, removing one of the vectors from the set would still let is span \( \mathbb{R^2} \).
The set \(B \) does form a basis for \( \mathbb{R^2} \) because it spans \( \mathbb{R^2} \) using the minimum number of vectors to do so, they are linearly independent.
When we write a vector as a column of scalars the vector is being expressed as a linear combination of some basis for that vector space. When no basis is explicitly stated we assume it is the standard basis being used.
Definition:
The standard basis for \( \mathbb{R^3} \) is \( \overrightarrow{i},\overrightarrow{j}, \overrightarrow{k} \) where
\( \overrightarrow{i} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \)
\( \overrightarrow{j} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \)
\( \overrightarrow{k} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \)
The standard basis for \( \mathbb{R^n} \) is the basis composed entirely of unit vectors
\( \overrightarrow{e_1} = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \)
\( \overrightarrow{e_2} = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix} \)
\( \overrightarrow{e_n} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix} \)
Consider the vector \( \overrightarrow{c} \),because the standard basis for \( \mathbb{R^2} \) is \( \overrightarrow{i} \) and \( \overrightarrow{j} \) the vector \( \overrightarrow{c} \) is a linear combination of \( \overrightarrow{i} \) and \( \overrightarrow{j} \).
Definition:
If \(S = \{ \overrightarrow{v_1}, \overrightarrow{v_2} ,\dotsc, \overrightarrow{v_n} \} \)is a basis for a vector space V, and
\( \overrightarrow{v} = c_1v_1 + c_2v_2 + \dotsc + c_nv_n \)
is the expression for a vector \( \overrightarrow{v} \) in terms of the basis S, then the scalars \( c_1, c_2, \dotsc, c_n \)
are called the coordinates of \( \overrightarrow{v} \) relative to the basis S. The vector
\( \overrightarrow{c} = \begin{bmatrix} c_1 && c_2 && \cdots && c_n \end{bmatrix} \) in \( \mathbb{R^n} \) constructed from these
coordiantes is called the coordinate vector v relative to S; it is denoted by
\( (\overrightarrow{v})_S = \begin{bmatrix} c_1 && c_2 && \cdots && c_n \end{bmatrix} \)
The standard basis vectors \( \overrightarrow{i} \) and \( \overrightarrow{j} \) are not the only possible basis vectors for \( \mathbb{R^2} \),in fact for any vector space there are an infinite number of sets of vectors which form a valid basis. Remember a basis is any linearly independent set of vectors which span a vector space.
Because vectors when expressed as a column of scalars describe a linear combination of their basis, choosing a different basis changes where the vector will land geometrically. Two vectors can be written with the same scalar components but look geometrically completely different if they written with respect to different basis.
In the interactive program below you can edit the four vectors \( {\color{vred}\overrightarrow{b_1}}, {\color{vgreen}\overrightarrow{b_2}}, {\color{vblue}\overrightarrow{b_3}}, \overrightarrow{c} \). The vector \( \overrightarrow{c} \) is a coordinate vector relative to the basis \( B = \{ {\color{vred}\overrightarrow{b_1}}, {\color{vgreen}\overrightarrow{b_2}}, {\color{vblue}\overrightarrow{b_3}} \} \). Change the directions of the basis vectors to see how the vector \( \overrightarrow{c} \) is transformed. Notice that the vectors \( {\color{vred}\overrightarrow{b_1}}, {\color{vgreen}\overrightarrow{b_2}}, {\color{vblue}\overrightarrow{b_3}} \) form a basis for \( \mathbb{R^3} \) unless you make them linearly dependent.
\( B = \{ {\color{vred}\overrightarrow{b_1}}, {\color{vgreen}\overrightarrow{b_2}}, {\color{vblue}\overrightarrow{b_3}} \} \)
\( {\color{vred}\overrightarrow{b_1}} \)
\( {\color{vgreen}\overrightarrow{b_2}} \)
\( {\color{vblue}\overrightarrow{b_3}} \)
\( \overrightarrow{c} = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} \)
\( \overrightarrow{c} = c_1{\color{vred}\overrightarrow{b_1}} + c_2{\color{vgreen}\overrightarrow{b_2}} + c_3{\color{vblue}\overrightarrow{b_3}}\)