Our final fact about nonsingular matrices expresses the correspondence between invertible matrices and invertible linear transformations. As a Sage demonstration, we will begin with an invertible linear transformation and examine two matrix representations. We will create the linear transformation with nonstandard bases and then compute its representation relative to standard bases.
We can convert T to a new representation using standard bases for QQ^4 by computing images of the standard basis.
Understand that any matrix representation of T_symbolic will have an invertible matrix representation, no matter which bases are used. If you look at the matrix representation of T_standard and the definition of T_symbolic the construction of this example will be transparent, especially if you know the random matrix constructor,