If v is a vector in a space V, once a basis B has been chosen, one may associate to v a column vector. The book simply calls this M(v), for the matrix of v. In class, we will often keep track of the basis explicitly, since the matrix associated to v depends greatly on B. Consequently, in class we write [v]_B.

Similarly, given a linear transformation f&isin L(W,V), the book simply calls the associated matrix M(f). Again, since this depends on the basis B for V and the basis C for W, we denote this [f]_{B &larr C}. Hulpke uses the notation _B[f]_C, which is also a reasonable solution. Note the direction of the letters; the inputs to f are elements written with respect to C, and the outputs are elements written with respect to B.