# Kernels and Images

October 2, 2011 Leave a comment

Let’s take a break from matrix representations, and instead think about linear transformations. For the rest of this post, let be a linear transformation. We are going to define images and kernels. Images are slightly easier to understand, so we’ll talk about them first.

##### Images:

The image of is a subspace of the codomain . In english, it means any vector we can get as a result of applying to something. In math, it is defined by:

Sometimes we want to talk about the dimension of the image, and want a shorthand for that. This is called the **rank** of the map, and is written:

.

##### Kernels:

The kernel of is a subspace of the domain . In english it means all of the things that sends to zero. In math, it is defined by:

Sometimes we want to talk about the dimension of the kernel, and want a shorthand for that. This is called the **nullity** of the map, and is written:

##### Some things:

You’ll notice that I’m being careless, and just writing when I should be writing . I’m also just saying let be a vector space, and not saying over what field we are working. Such is mathematics. When things are clear, we don’t like to write out all the details. Linear transformations *must* be between vector spaces over the same field, and the dimension of a vector space is the dimension over the implied field. I’ll write it all down whenever it isn’t clear.

I personally prefer to say and instead of nullity and rank, you should know the terms. Tomorrow we’ll prove a good theorem relating these things.