MHB Linear Algebra: Kernel & Range of Linear Transformation

[matqkks]

Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?

 

[Siron]

They can be useful. Suppose you have a linear map f: V \to W. If you want to know if this linear map is injective (i.e one-to-one map) then you can take a look at the kernel: $$\ker( f)=\{0\} \Leftrightarrow \ f \ \mbox{is injective}$$

There's also the following result.
$$V/\mbox{ker}( f) \cong \mbox{Im}( f)$$
which can be very useful because it's easier to work with the image in stead of the quotientspace $V/\mbox{ker}( f)$

These are offcourse a lot of other results but these two are the first I could remember immediately.