Question

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

Short answer

The $B$ matrix for the identity transformation $I:V\to W$ is $\left(e_1...e_n\right)$.

Step-by-step solution

Use the given information 

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 ${\mathbb{R}}^n$.

Find the B matrix 

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:

$\begin{array}{rl}{\left(I\left({\mathrm{b}}_j\right)\right)}_B& ={\left(b_j\right)}_B\\ & =e_j\end{array}$

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

$\begin{array}{rl}{\left(I\right)}_B& =\left({\left(I\left(b_1\right)\right)}_B........{\left(I\left(b_n\right)\right)}_B\right)\\ & =\left(e_1...e_n\right)\\ & =I\end{array}$

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