Vector Space Isomorphism

up:: Linear Transformation

A linear transformation $T:U\to V$ 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 $v_1,v_2\in V$. Then we have that $\exists !u_1,u_2\in U\mid T(u_i)=v_i$. Thus

$$\begin{array}{rl}{T}^{-1}(v_1+\lambda v_2)& =T^{-1}(T(u_1)+\lambda T(u_2))\\ & =T^{-1}(T(u_1+\lambda u_2))\\ & =u_1+\lambda u_2\\ & =T^{-1}(v_1)+\lambda T^{-1}(v_2)\end{array}$$

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),

$$\dim U=\dim ImT=\dim V$$