7.4, or the section at which I realized that summarizing 3 of these a day really sucks.

Section 7.4 is entitled “Homogeneous and Inhomogeneous Systems.”

Just as a reminder, a homogeneous system has the form Ax = 0. Obviously, if we picked x to be the zero vector, then we would have a solution, since A0 = 0. Then we will call any solution x that isn’t 0 a nontrivial solution. So to be nontrivial, the vector must contain at least one nonzero component.

So, when you’re solving the system Ax = 0, you will go and augment A and 0 and bring it to row echelon form (or row reduced, either one). You will notice that the last column of your row echelon form will contain only zeroes. In other words, it didn’t change from the original zero vector it was. It’s not too difficult to see that this will always be the case. So it’s not really necessary to augment the coefficient matrix with the zero vector, but I suppose if you wanted to, I can’t stop you.

“The homogeneous system Ax = 0 has a nontrivial solution if and only if a matrix in row echelon form that is equivalent to A has a free column” (302).

“Any homogeneous linear system with fewer equations than unknowns has a nontrivial solution” (302).

So with inhomogeneous systems, our solution set will look vastly similar to stuff in section 7.2. We’ll have a particular solution to the inhomogeneous system Ax = b. We’ll denote this particular solution with a p. Our solution set for this system will have the form x = p + v, where Av = 0.

In other words, we’ll have the homogeneous solution (if the system was homogeneous, that is) and a particular solution together for the solution set for our inhomogeneous system.

This makes the homogeneous solution set very important, since we use it both for homogeneous and inhomogeneous solution sets. It’s so important, in fact, that it gets a special name. The nullspace of A is the set of all solutions to the homogeneous equation Ax = 0. It’s denoted by null(A).

A theorem concerning nullspace:

“The solution set of the inhomogeneous system Ax = b has the form

where p is any particular solution to the system Ax = b” (305).

Also, there are two properties of nullspace which are relevant:

1. If x and y are vectors in null(A), then x + y is also in null(A).

2. If x is in null(A) and a is a number, then ax is also in null(A).

Here’s an example concerning nullspace of a matrix:

If we name the variable for column one x₁ and for column two x₂, then x₂ is a free variable and we will give it the value x₂ = t. So then if we solved for x₁, then we would get x₁ = -5x₂.

Our variable t is just any arbitrary number of your choosing.

One final thing to leave you with for 7.4: a unit vector is a vector that has length one.

All right, that’s it for section 7.4. Onto 7.5!