Ytukyg

The system:
$$ T(x[n]) = ax[n] + bx[n-3] $$

For me it seems that the system is linear:

$$
begin{align}
T(alpha_1x_1[n]+alpha_2x_2[n]) & = a(alpha_1x_1[n]+alpha_2x_2[n]) + b(alpha_1x_1[n-3]+alpha_2x_2[n-3]) \
& = alpha_1(ax_1[n] + bx_1[n-3]) + alpha_2(ax_2[n] + bx_2[n-3]) \
& = alpha_1T(x_1[n])+alpha_1T(x_2[n])
end{align}$$

Thus it's linear, however in the presentation I got it says it's not linear. (without reasoning) Where I'm making the mistake (or maybe there is a mistake in the presentation)

I believe there's either a mistake in the presentation or the presentation is using a different definition of linear.

For example, the system is linear in $x$ from a system perspective, but it's affine in $x[n]$ (and, therefore not linear) because of the $bx[n-3]$ offset.

On this site, we tend to go with the system definition rather than split hairs about linearity versus affine-ness.

[Note: it may happen that a teacher makes a oral mistake, that puzzles the audience. So here is an alternative explanation on this system being non-something]

This system is, as far as Peter K., Matt L. and I know, nicely linear. You already did the computations. With a little more work, among classical properties, it is also time-invariant, causal, stable.

The only basic property it does not possess is "to be memoryless", unless one of the constant $a$ or $b$ (including both) is zero (see for instance an extended conception of a memoryless system from What is a memory less system?).

A system $mathcal{T}$ is linear if its response to a weighted sum of two signals equals the weighted sum of its individual responses to those two input signals:

$$mathcal{T}big{alpha x_1+beta x_2big}=alphamathcal{T}big{x_1big}+betamathcal{T}big{x_2big}tag{1}$$

with arbitrary constants $alpha$ and $beta$, and arbitrary input sequences $x_1[n]$ and $x_2[n]$. A system $mathcal{T}$ satisfying $(1)$ is completely characterized by its impulse response $h[n]$, and its input-output relation can be formulated as a convolution of the input sequence with its impulse response:

$$mathcal{T}big{xbig}=sum_{k=-infty}^{infty}h[k]x[n-k]tag{2}$$

It is easily shown that the given system

$$mathcal{T}big{xbig}=y[n]=ax[n]+bx[n-3]tag{3}$$

satisfies $(1)$, and that it is characterized by the impulse response

$$h[n]=adelta[n]+bdelta[n-3]tag{4}$$

Clearly, its input-output relation $(3)$ can be written as a convolution sum $(2)$. Equivalently, the $mathcal{Z}$-transform of its response is given by the multiplication of the $mathcal{Z}$-transform of its input sequence and its transfer function $H(z)=mathcal{Z}{h[n]}$:

$$Y(z)=X(z)H(z)tag{5}$$

By contrast, an affine system does not satisfy $(1)$, and it cannot be characterized by an impulse response or, equivalently, by a transfer function. The given linear system $(3)$ could be made affine by adding a constant $c$ ($cneq 0$) to its output:

$$mathcal{T}big{xbig}=y[n]=ax[n]+bx[n-3]+ctag{6}$$

This input-output relation cannot be formulated in terms of a convolution $(2)$, and it can easily be checked that the system $(6)$ doesn't satisfy $(1)$. Such a system is not linear, since part of the output (the constant $c$) does not depend on the input signal $x[n]$.

There is no reasonable definition of linearity according to which the given system $(3)$ could be classified as being non-linear.

This system is, as far as Peter K., Laurent, and I know, nicely linear. It is also time-invariant, causal, and stable. Indeed it's LTI with impulse response $h[n] = a delta[n] + b delta[n-3]$.

The only basic property it does not possess is "to be memoryless", unless $a$ or $b$ is zero (see for instance an extended version from What is a memory less system?).