Can linear transforms of vector space be seen as LTI systems?
The question of abstract nature so a rigorous answer could require more elaboration, however on the surface one can at least observe the following.

An LTI system is also a linear transform, which maps its input signal (vector) $x(t)$ or ($ x[n]$) into an output signal (vector) $y(t)$ (or $y[n]$), where both the input and output belong to the same signal space, one kind of vector space.

For such transforms and vector spaces the time invariance property can be checked easily such as $y(t-d) = \mathcal{T} \{x(t-d) \}$, but for other vector spaces such as the n-tuples or matrix spaces etc, the operation of time invariance is not trivial to check, at least intuitively. The shift of a vector (an element of the set) should be defined first.

May be the following links could help further:

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