Vector Space Isomorphism

up:: Linear Transformation

A linear transformation is said to be an isomorphism if it is Bijective.

Properties

• Every vector space with same dimension are isomorphic to each other
• A linear transformation between spaces of equal dimension is injective iff it’s surjective iff it’s an isomorphism

A bijective linear transformation already has linear inverse:
Let . Then we have that . Thus

Isomorphisms preserve dimension:
Via the Kernel-Image Theorem, we have that, since A linear transformation is injective iff its kernel is trivial, as well as being Surjective (Image equals codomain),