Kernels and Images

Let’s take a break from matrix representations, and instead think about linear transformations. For the rest of this post, let T:V\to W 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 T is a subspace of the codomain W. In english, it means any vector we can get as a result of applying T to something. In math, it is defined by:

\mbox{Im }T=\{Tv\mid v\in V\}

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:

\mbox{rank }T=\dim\mbox{Im }T.

Kernels:

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

\ker T=\{v\in V\mid Tv=0_W\}

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:

\mbox{null }T=\dim\ker T

Some things:

You’ll notice that I’m being careless, and just writing \dim when I should be writing \dim_k. I’m also just saying let V 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 \dim\ker T and \dim\mbox{Im }T instead of nullity and rank, you should know the terms. Tomorrow we’ll prove a good theorem relating these things.

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