Image and Kernel

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.

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13 thoughts on “Image and Kernel

  1. Your haircut looks great, if you were wondering. Other than that, I got no idea what this is about, I guess I will have to read up on it 😁 👍✌

  2. Love your videos. Nice haircut btw. Looks great. Can you please make a few videos on calculus too

  3. Quick question: If I understand kernels correctly, then that means that the Dimension of them will generally be 1?

    And the Dimension of an Image will generally be what it is equal to — so if it equals ax^2+bx for example, then the dimension will be 2?

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