Is this function linear in x?

Question

Is the function

$$f(x,t) = e^{t} \cdot x$$

considered linear in x? The exponential term is throwing me off a bit, in terms of definitions, but it would seem that it is linear in x, since the power of x is 1. Just wanted to be sure - thanks.

Answer 1

Studying with respect to $x$, observe that $e^t$ is nothing more than a constant generated by different values of $t$. By linear map definitions :

$$f(x+y,t) = (x+y)e^t = xe^t + ye^t = f(x,t) + f(y,t) $$

$$f(cx,t) = e^t \cdot c \cdot x = c \cdot e^t \cdot x = cf(x,t)$$

Thus the function $f(x,t) = e^t \cdot x$ is linear with respect to $x$.