Kernel and Image

Time to define some new terms!!!!

Let’s lay ’em down?!

Here we go~~~~

First, picture a function $L$ like this:

The “kernel of $L$” — written ker L — is

this thing right here!!!!

The set of all $v$ in $V$ that $L$ sends straight to zero!!!!

To make it concrete, here’s a baby example:

And one more thing!!

Don’t forget — ker L lives inside $V$. It’s a subset of $V$.

Next up: the “image of $L$”, written im L.

Definition? Here:

im L is, basically… you remember that $W$ from before? It’s “the set of targets that actually got hit by arrows.” That.

So — Range of $L$. The reachable territory, the stuff $L$ can actually output. Sometimes you’ll see it written Range or R(L) instead.

And don’t miss this: im L is a subset of $W$!!!!!

Now, if $L$ is injective~~~

Or if $L$ is surjective~~~

Here’s how that shakes out..

And the theorem you absolutely cannot forget — the one from earlier:

When $V$ and $W$ in $L : V \to W$ have the same dimension,

①: $L$ surjective → injective conditions are automatically satisfied → so $L$ is bijective.

②: $L$ injective → surjective conditions are automatically satisfied → so $L$ is bijective.

Burn this one in too!!!

Originally written in Korean on my Naver blog (2016-01). Translated to English for gdpark.blog.