Isomorphism

An isomorphism is a Linear Map $T:V\to W$ that is:

• Injective
• Surjective or:
• Has an Inverse Function

Isomorphic Definition

If $V,W$ are isomorphic, then there exists an isomorphism $T:V\to W$

Isomorphism Formal Definition

A linear transformation $T:V\to W$ is an isomorphism $⟺$ $T$ is Injective and $T$ is Surjective.

Formal Definition for $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$

A function $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is isomorphic if it sends the unit square to a paralellogram with non-zero signed area. $[0,1]\times [0,1]=\{a{e}_1+b{e}_2:a,b\in [0,1]\}$ So, if you calculate a non-zero Determinant, the function is isomorphic.

Concepts

Proving Isomorphism

Example

Prove $M_{2\times 2}(\mathbb{R})\to {\mathbb{R}}^4$ is isomorphic

1. Consider $T:M_{2\times 2}(\mathbb{R})\to {\mathbb{R}}^4$ as $T([abcd])=(a,b,c,d)$
2. Note that $T$ is a linear map
3. Moreover, it is surjective and injective
4. Therefore, it is an isomorphism by definition

Example 2

Prove $P_3(\mathbb{R})\to {\mathbb{R}}^4$ is isomorphic

1. Consider $S(a{x}^3+b{x}^2+cx+d)=(a,b,c,d)$
2. Note $S$ is linear, injective and surjective
3. Therefore $T$ is an isomorphism by definition