Assignment: Find the number of parameters in the general solution to a system of linear equations

Question

This is a question given in an assignment I'm working on:

If the coefficient matrix $A$ in a homogeneous system of 33 equations with 28 unknowns is known to have rank 12, how many parameters are there in the general solution?

I've deduced that, since this is a homogeneous system with fewer variables than equations, the only solution is the trivial solution; I'm unsure, however, how to find the number of parameters in this solution.

How would I go about finding the number of parameters in this kind of abstract situation?

Answer 1

You add a parameter for every column without a pivot in the REF form of the coefficient matrix. That is, there are the same number of parameters as the dimension of the null space of $A$. What is the relationship between $\dim NS(A)$ and $\text{rk}(A)$?

Mouse over for relationship...

$$\dim NS(A) = \text{columns} - \text{rk}(A)$$