the Nature of Linear Transformation

what's a linear function? (aka linear map, linear transformation.)

by definition, it's a function f on vector space such that:

f[x + y] == f[x] + f[y]
f[α x] == α f[x]

where α is a scalar.

but what does it mean intuitively?

• The zero vector is never moved.
• The transform is linear in nature. Meaning, kinda uniform thru-out, as in uniform scaling. It includes isometries of {rotation, reflection, scaling} but does not include translation (because that moves the zero vector)