Notes on typography:

Square brackets [  ] around matrices, including column vectors, are generally omitted here.

The notation "V_k" is used to indicate "V" with subscript "k". For example, if there are three variables in a linear equations problem they will often be denoted x_1, x_2, x_3. The notation "a^3" is used to indicate "a" with the superscript 3. E.g., R^2 indicates the vector space of 2 by 1 real column vectors. We use "u.v" to indicate the dot product of the vectors u and v. In characteristic polynomial problems we use x instead of lambda.