Given Information
We are given with vectors $\mathbf { u } _ { 1 } , \ldots , \mathbf { u } _ { p } ,$ and $\mathbf { w }$ in $V ,$. We have to show that $\mathbf { w }$ is a linear combination of $\mathbf { u } _ { 1 } , \ldots , \mathbf { u } _ { p }$ if and only if $[ \mathbf { w } ] _ { \mathcal { B } }$ is a linear combination of the coordinate vectors $\left[ \mathbf { u } _ { 1 } \right] _ { \mathcal { B } } , \ldots , \left[ \mathbf { u } _ { p } \right] _ { \mathcal { B } }$

Step-1:
let us assume that $\mathbf { w }$ is the linear combination of $\mathbf { u } _ { 1 } , \mathbf { u } _ { 2 } , \ldots , \mathbf { u } _ { p }$. then, \[ \mathbf { w } = c _ { 1 } \mathbf { u } _ { 1 } + c _ { 2 } \mathbf { u } _ { 2 } + \cdots + c _ { p } \mathbf { u } _ { p } \] We need to prove that \[ \left[ \mathbf { u } _ { 1 } \right] _ { E } , \left[ \mathbf { u } _ { 2 } \right] _ { B } , \ldots , \left[ \mathbf { u } _ { p } \right] _ { B } \] i.e. $[ \mathrm { w } ] _ { B }$ is a linear combination of the coordinate vectors

Step-2:
Let $\mathcal { B } = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \ldots , \mathbf { b } _ { n } \right\}$ be the basis for a vector space $V .$ Then the coordinate mapping $\mathbf { x } \mapsto [ \mathbf { x } ] _ { \varepsilon }$ is one-to- one linear transformation from $V$ onto $\mathbb { R } ^ { n }$ . Then, \[\begin{array}{l} {[{\bf{w}}]_B} = {\left[ {{c_1}{{\bf{u}}_1} + {c_2}{{\bf{u}}_2} + \cdots + {c_p}{{\bf{u}}_p}} \right]_B}\\ = {\left[ {{c_1}{{\bf{u}}_1}} \right]_{\cal B}} + {\left[ {{c_2}{{\bf{u}}_2}} \right]_{\cal B}} + \cdots + {\left[ {{c_p}{{\bf{u}}_p}} \right]_{\cal B}}\\ = {c_1}{\left[ {{{\bf{u}}_1}} \right]_B} + {c_2}{\left[ {{{\bf{u}}_2}} \right]_B} + \cdots + {c_p}{\left[ {{{\bf{u}}_p}} \right]_B} \end{array}\] Therefore,

$[ \mathrm { w } ] _ { B }$ is a linear combination of the coordinate vectors $\left[ \mathbf { u } _ { 1 } \right] _ { B } , \left[ \mathbf { u } _ { 2 } \right] _ { B } , \ldots , \left[ \mathbf { u } _ { p } \right] _ { B }$