Part 1:

Part 2:

Part 3:

Definition/Proposition - Let \(V\) be a vector space and \(B = \mathbf{v}_1,\ldots, \mathbf{v}_n\) be a basis for \(V\). Let \(\mathbf{v} \in V\). Then there exists a unique representation \(\mathbf{v} = c_1\mathbf{v}_1 + \cdots + c_n\mathbf{v}_n\) where \(c_1,\ldots, c_n\in k\). Then define the coordinate vector of \(\mathbf{v}\) relative to \(B\), denoted \([\mathbf{v}]_B\), by \([\mathbf{v}]_B = \left[\begin{array}{c} c_1\\ c_2\\ \vdots\\ c_n\end{array}\right].\)