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?

…“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 $L : V \to W$ are finite-dimensional and 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!!!

Heads-up. The figures here are the original notes, slips and all. A few things to watch for:

• The baby example lists which of $v_1, \dots, v_4$ get sent to zero. Since $L$ is linear, the kernel also holds the zero vector and every combination $a v_1 + b v_2$: it is a whole subspace, not just the two vectors the picture writes.