Chapter 6.3, Problem 53E

True or False? In Exercises 53 and 54, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.

(a) If $T:R^n\to R^m$ is a linear transformation such that

$\begin{array}{l}T\left(e_1\right)={\left[a_{11},a_{21}\dots a_{m1}\right]}^T\\ T\left(e_2\right)={\left[a_{12},a_{22}\dots a_{m2}\right]}^T\\ ⋮\\ T\left(e_n\right)={\left[a_{1n},a_{2n}\dots a_{mn}\right]}^T\end{array}$

then the $m\times n$ matrix $A=\left[a_{ij}\right]$ whose columns corresponds to $T\left(e_i\right)$ is such that $T\left(v\right)=Av$ for every $v$ in $R^n$ is called the standard matrix for $T$.

(b) All linear transformations $T$ have a unique inverse $T^{-1}$.