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.
Are there known conditions under which a Markov Chain is also a Martingale? I know only that the only Random Walk that is a Martingale is the symmetric one, i.e., p= 1-p =1/2.
Hello !
I derived equations of stress tensor 2D transformation.
Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture.
I want to obtain expression that connects tensor for case 1 and tensor for case 2.
My attempt:
Are these equations correct? Is there more easier expression for stress tensor...