Professor Dave again, let’s talk image and kernel.
Now that we have learned about linear transformations, we have to discuss two related concepts, and
these are called image and kernel.
These are best defined by example, so let’s take a look at one now.
Say we have a linear transformation that maps from the vector space V to the vector space
W. As we know, this will involve taking vectors from V and turning them into vectors in W.
If we transform a group of vectors from V, we end up starting to map out several vectors
in W. This is the idea behind what we mean by image.
If we take a subspace of V, let’s call it S, this is a group of vectors from V that
can then be transformed.
The set of vectors that we can get from this transformation is what is known as the image
of S. One way to think of it is that it’s as if we are shining a light on a part of
the vector space V and seeing how much of W gets lit up.
The area of W that gets lit up is our image.
The image of the entire vector space V, denoted this way, has a special name, it is called
the “range of L”.
For example, consider the linear transformation that maps R3 to R2 given by L(v) = (v1, v2
- v3) for any vector v in R3.
For this example let’s say our subspace of V is given by the vectors of length 3 where
the second element is two times the first element, and the third element is 0.
Written out, this is the set of vectors of the form (c, 2c, 0) where c is any scalar.
Let’s plug this form into our linear transformation and see what we get.
Our transformation says to plug the first element into the top row, so c, then plug
the difference of the second and third element into the bottom row, so 2c minus 0.
We end up getting an image of the form (c, 2c).
The image of our subspace is the set of vectors of length two where the second element is
twice the first.
Now let’s move on to kernel.
Keeping in mind the fact that our linear transformation maps from V to W, the kernel of L, denoted
as shown, is the set of vectors in V that when transformed become the zero vector in
W. To keep zero vectors distinct, we will write the zero vector from V as zero sub V,
and the zero vector from W as zero sub W. Let’s take our linear transformation from
earlier for an example.
We have the mapping from R3 to R2 given by L(v) = (v1, v2 - v3).
To find the kernel, we want to find which vectors in R3 give 0W under this transformation.
This just ends up boiling down to solving the equation L(v) = 0W for possible values
of v1, v2, and v3.
In this case, that will be as follows: (v1, v2 - v3) = (0, 0).
This reduces to a simple pair of equations: v1 = 0 and v2 - v3 = 0.
Solving these gives us v1 = 0 and v2 = v3.
This means that any vector from R3 of the form (0, c, c) where the first element is
0 and the second and third are equal to one another, will give the zero vector in W when
transformed.
The set of vectors of this form is our kernel.
Before we move on, it’s worth noting that the kernel of L is a subspace of V, and for
any subspace, S, of V, the image of S is a subspace of W. So any vectors from a kernel
or image immediately follow all the properties of subspaces we already learned.
Now with these definitions understood, let’s check comprehension.