exerpt from Link: Isomorphisms of Vector Spaces
The word derives from the Greek iso, meaning “equal,” and morphosis, meaning “to form” or “to shape.”
they are connected by an invertible linear transformation (mapping) between two sets (vector spaces), which preserves the addition and scalar multiplication operations among elements:
Consider the logarithm function: For any fixed base b, the logarithm function maps from the positive real numbers R+ onto the real numbers R; formally:
This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group (R+,×) of positive real numbers under ordinary multiplication. The logarithm function obeys the following identity:
But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group (R+,×) to the group (R,+).