Question about linear combination of non-stationary signals Suppose $f_1,\ f_2,\ ...\ f_{n-1},\ f_n$ are $n$ non-stationary discrete signals of same length and linearly independent. Each $f_i$ is an intrinsic mode function (IMF) of a seismic signal evaluated by empirical mode decomposition (EMD). That is, they are amplitude-modulated, frequency-modulated signals. Linear combination of these signals can be expressed as follow:

$$F(t) = \sum_{j=1}^{j=n} \alpha _j f_j(t)$$

where, $\alpha_js$ are real numbers and they are to be selected such that $\sum_{j=0}^{j=n} \alpha_j\ne0$. Is it true to claim that $F(t)$ is also a non-stationary signal?
No, try with $\alpha_j = 0$, $\forall j$.

I suspect this to be a tricky question, to deceive fast thinking. For instance, having the "same length" does not mean a lot, for a continuous variable $t$. Misleading fake clues to make the question look serious.

Without more hypotheses on the notion of stationarity (in law, wide-sense) and the statistical interrelations between the so-called signals (it would be wiser to talk about processes). See Marcus' answer for more examples.