In this example, we are going to investigate what is called the Null Space of a matrix.
So the particular matrix that will work within this problem is the matrix A and it is defined
right here as a 3 by 3 matrix.
And we are going to be interested in determining if certain vectors are in the null space of
A or not in the null space of A. So in the first part here, in part A, we were
given the vector x which is the 3 by 1 vector 1, 1, 2, and we are asked is this vector in
the null space of A?
So matrices have no spaces and those no spaces consist of all the vectors such that Ax equals
the all zero vector.
So if I can find a vector x such that Ax equals 0, 0, 0, in this case, then x is in the null
space.
And if x is not equal to the all zero vector, then it is not in the null space.
So when you're given a matrix and you're given a vector in and asked to check is this factor
in the null space?
That's a pretty simple thing to do.
You really just need to form the matrix vector product Ax and compute to see what it's equal
to.
So in this case, we write down the matrix A. We write down the vector x.
We perform the matrix multiplication.
So I'm going to get 1 times that, negative 2, 6, negative 10, and a negative 4, 3, 8.
So after I form the product A times x, I end up with this vector right here.
So you can see what I did.
I just did my normal, this row times this linear combination to get minus 2, minus 9,
minus 10.
And then 2, negative 6, 5 times this to get 2, negative 6, 10.
And then, finally, this times this to get the final row.
So if I add all these up, a negative 2, minus 9, minus 2 is equal to negative 13.
2 minus, 6 minus, 10 is equal to negative 14.
And minus 4, minus 3 plus 8 is equal to 1.
So after computing A times x, I end up with the vector and negative 13, and negative 14,
1, which is obviously, not equal to 0, 0, 0.
So this is after that we were given 1, 1, 2 is not in the null space because I did not
end up with the vector 0, 0, 0.
That was pretty easy.
Let's do another quick example.
Let's check to see if this vector is in the null space of A.
So again, our approach is exactly the same to figure out if a given vector is in a null
space.
All we need to do is take A times x.
And again, if I do that, I'm going to end up with first, that row times that column
element lines, would be a negative 26 plus 36 minus 10.
And then, the next entry will be this row times that, so I'll end up with 26 plus 24
minus 50.
And then, finally, the last entering the vector will be that row times that column, element-wise--
so negative 52 plus 12 plus 40.
And you can see what happens here if I add these up now.
A negative 26 plus 36 minus 10 is really a negative 36 plus 36, which is zero.
And then, similarly, here 26 plus 24 is 50, 50 minus 50 is 0.
And then finally, right here I have a negative 52 plus 12 plus 40.
12 plus 40 is 52 and negative 52 plus 52, is 0.
And I end up with the all zero vector.
So I took the matrix A multiply that by x.
I ended up with the all zero vector.
So by definition of the null space, that means that this vector x must be in the null space
of A. So that's the end of this example.
We now know how to check if for a given matrix and a given vector, if the null if the matrix
is in the null space and all you have to do is multiply A times x.
If you don't get the all zero vector, it is not in the null space.
And if you get the all zero vector, it is in the null space.