linear space

1. a basis of linear space $V^k$,which is $x_1,x_2,...x_k$,can be called as a coordinate system.let vector $v\in V^k$ and it can be linear expressed on this basis as $v=a_1{x}_1+a_2{x}_2+...+a_k{x}_k$,the $a_1,a_2,....,a_k$ is coordinate in this coordinate system denoted by $(a_1,a_2,...,a_k{)}^T$.
2. the various coordinate systems for the same vector are different usually because of non-uniqueness of basis of a linear space. for the first basis which is $x_1,x_2,...x_k$ ,the coordinate is $(a_1,a_2,...,a_k{)}^T$ and there are the second basis $x_1^{\prime },x_2^{\prime },...x_k^{\prime }$ to coorespond another coordinate $(a_1^{\prime },a_2^{\prime },...,a_k^{\prime }{)}^T$,also can be explain that $v=a_1{x}_1+a_2{x}_2+...+a_k{x}_k=a_1^{\prime }{x}_1^{\prime }+a_2^{\prime }{x}_2^{\prime }+...+a_k^{\prime }{x}_k^{\prime }$.
3. let $v\in V^k$ and $x_1,x_2,...x_k$ is a basis of linear space,then $v$ can uniquely be separated into the linear combination that $v=a_1{x}_1+a_2{x}_2+...+a_k{x}_k$.

references

1. deepseek
2. 《矩阵论》