For the null space of B.

And this is our null space.

This guy's null space is going to be the null space of that guy in reduced row echelon form.

That's the definition of the null space.

It's the dimension of the null space.

The null space of the reduced row echelon form is the same as the null space of our original matrix.

So the null space of B was equal to the null space of the reduced row echelon form of B.

So your null space will always contain that.

So the null space of this guy we could just put in reduced row echelon form the null space of this guy is the same thing as the null space of this guy in reduced row echelon form.

So we can write that the null space of A is equal to the null space of the reduced row echelon form of A.

If your null space is trivial, what does that mean?

The solution set is just the null space of r.

So the null space of this matrix is the eigenspace.

Plus some homogeneous solution, so some member of the null space.

Your null space will always, at minimum, contain the 0 vector.

So this will be some special member of your null space.

Well the solution set of this is just the null space.

These guys form a basis for the null space of B.

What is that the dimension of my null space of B?

So the dimension of the null space of B is 3.

In this case, we'll calculate the null space of matrix A.

And only non invertible matrices have a non trivial null space.

They're the exact same null space, the exact same solution set.

Where this guy right here is a member of the null space.

That we know that they span the null space, but if they're

So v1, the set v1, v2, and v3 is actually a basis for the null space, for the null space of Oh, you know what, I have to be very careful.

What does it mean if your null space only contains a trivial vector?

We call this right here, we call n, the null space of a.

But in this video let's actually calculate the null space for a matrix.

So what is the null space of this is the same thing as the null space of this in reduced row echelon form So let's put it in reduced row echelon form.

By definition our null space is all of the x's that satisfy this equation.

The null space is just the 0 vector, this part of your solution disappears.

And this is going to be true of any null space problem we do.

We spent a good deal of time on the idea of a null space.

And then we go off and we figure out the null space of A.

That means that the null space of this matrix has got to be nontrivial.

So, since its a member of our null space, we can say that it is equal to so our any solution minus our particular solution is equal to some member of our null space.

But this is what, essentially, your null space is spanned by these vectors right there.

Our orange n is equal to the notation is just the null space of a.

So here are all of the x's, this is the null space, this problem, if we find all of the x's here, this is the null space of the reduced row echelon form of our matrix A.

If your null space contains the 0 vector, then all of your columns are linearly independent.

We find out that the null space of A contains more than just the zero vector.

If its null space only has the 0 vector, then that means its columns are linearly independent.

If all of these guys are linearly independent, then the null space is just the 0 vector.

So this is the null space of a, is all of the x's that satisfy this equation.