I'm afraid you don't understand convolution
[1 2 3 4]
decomposed signals are now
[1 0 0 0]
[0 2 0 0]
[0 0 3 0]
[0 0 0 4]
If you were to individually apply each to the system, you'd end up with
[0 2 4]
[0 0 3 6]
[0 0 0 4 8]
Now a linear combination of these (which here is element-wise addition) yields
[1 4 7 10 8]
This is what your convolution equation gives you as well.
If I have my $g(t)$, can I decompose it in different ways?
Any way you like !
Could we write $g(t)$ above as a convolution of
[1 4 7 10 8], with
If yes, do you see how your decomposed signals are now different ?
Fourier transform does something similar as well, but, it decomposes signals into sinusoids.
If yes what are those few decomposed signal
Try to work out the answer above.
Being helpful (since you're starting off with DSP), there is usually a good depth in definitions and a quite some detail in wordings. Try to understand each in their appropriate context to get the complete picture !
Follow this OCW and good luck !
Disclaimer : I've liberally used terms like
linear. Like @Stanley points, they are very broad and only a narrow definition is used here.