All linear operators map origin to origin. But translation applied to the origin will shift it. So translation cannot be linear.

Let's take another approach.

Take a point p that's sum of vectors a and b, that is

p = a + b.

Now, if translation was a linear transformation, then translating p (say along x-axis by 1 unit) is equivalent to applying same translation to a and b separately and then summing them. But the latter ends up translating by twice the amount. Or in other words

p +t ≠ (a +t) + (b +t) = p + 2t.

So translation is not a linear operators in this vector space.