Let \(V\) be a vector space with a basis \(B = \left\{ {{b_1},........{b_n}} \right\}\). Find the \(B\) matrix for the identity transformation\(I:V \to W\).

It is given that \(I:V \to W\)and v be a vector space in \({\mathbb{R}^n}\). So, the coordinate vector for each \({j^{th}}\) vector of the basis \(B\) is \({e_j}\), which is a standard basis vector for \({\mathbb{R}^n}\).

For example, the vector in the basis \(B\), is a linear transformation of the identity matrix \(1 \times n\), \(I = \left( {1\,,0,\,0.....,\,0} \right)\) , as shown follows:

\({b_1} = 1 \cdot {b_1} + 0 \cdot {b_2} + ...... + 0 \cdot {b_n}\)

Thus, the identity transformation \(I\) relative to basis \(B\) can be done as follows:

\(\begin{aligned}{}{\left( {I\left( {{{\rm{b}}_j}} \right)} \right)_B} &= {\left( {{b_j}} \right)_B}\\ &= {e_j}\end{aligned}\)

So, we can further simplify the matrix I to write it as follows:

\(\begin{aligned}{}{\left( I \right)_B} &= \left( {{{\left( {I\left( {{b_1}} \right)} \right)}_B}........{{\left( {I\left( {{b_n}} \right)} \right)}_B}} \right)\\ &= \left( {{e_1}\,...\,{e_n}\,} \right)\\ &= I\end{aligned}\)

Thus, the \(B\) matrix for the identity transformation \(I:V \to W\) is \(\left( {{e_1}\,...\,{e_n}\,} \right)\).