time linearity of LTI system There is this following question.
Consider the transformation H {x}[n] = n x [n]. Does H define an LTI system?
What I understood from the question is:
x[n] -&gt; H -&gt; y[n]
or according to the question,
x[n] -&gt; H -&gt; n x[n]
to prove the time invariance, if I put n-no instead of n, it should be,
x[n-no] -&gt; H -&gt; y[n-no] = n x[n-no]
but it turns out to be,
x[n-no] -&gt; H -&gt; n x[n - no] - no x[n-no]
which means the output depends on extra term no x[n-n0] and is not time invariant.
Incase, the trasformation was any constant * x[n] the it would have conserve the time invariance property, if I am not wrong.
But the proof mentioned in the notes is,

I have difficulty understanding the proof. If anyone could explain it to me or if I understood the question wrong, I'd appreciate it.
Your notes gave a counter example, you just proved it in the general case.
Your note used your x[n] as delta and showed that
y[n-1] unequal to H{x[n-1]} which is exactly what you showed.
In the notes example H{x}=0 for all n, as discrit delta equals zero foll all n but n=0, which means H is zero every where for all input of the kind delta[n-n0]  (becouae of the n factor which zeros y at its only non zero point)
Than you have zero output for delta input. But as you showed time varied output isn't zeros every where.