I would say no. Not if The vectors you have exhibited there are a basis. So, take any vector in Then where the hatted vectors are the unit vectors you mentioned. Then...does there exist a linear transformation where...

Therefore,

However, this is not the situation you're dealing with in the first part of your post. In the first part, the book says that the three vectors there aredependent. A basis is, by definition, independent. Therefore, all bets are off.