Regarding your example F(x,y, z) = (3x +y, y+z, 2x-3z), consider the following.
Let , and . Then .
Obviously or .
Standard canonical basis in , vectors , , are mapped to , columns of the matrix :
, and .
Every two bases in vector space V are equivalent. Meaning that every vector can be given in either one of them as a linear combination of the vectors forming the basis, the difference is in coefficients used.
If nothing is specified then standard basis is assumed.