3D Inevitability and Isomorphisms

Invertible Linear Maps

Definition

1. A linear map $T\in \mathcal{L}(V,W)$ is called invertible if there exists a linear map $S\in \mathcal{L}(W,V)$ (in the opposite direction) such that $ST$ equals the identity operator on $V$ and $TS$ equals the identity operator on $W$
2. A linear map $S\in \mathcal{L}(W,V)$ such that $TS=I$ and $ST=I$ is called the inverse of $T$.

The inverse is unique

An invertible linear map has an unique inverse Proof

1. Assume $S_1$ and $S_2$ are both inverses of $T$
2. then: $S_1=S_{1}I=S_1{S}_{2}T=S_{1}T{S}_2=I{S}_2=S_2$

Notation

The inverse of a map $T$ is denote $T^{-1}$

Example

Find the inverse of $T(x,y,z)=(-y,x,4z)$ The inverse of the map is $T^{-1}(x,y,z)=(y,-x,\frac{1}{4}z)$

Conditions

A linear map is invertible $⟺$ it is injective and surjective (bijective) Proof: #todo if you want to spend time on this proof later.

Injectivity is equivalent to surjectivity

Suppose that $V$ and $W$ are finite-dimensional vector spaces, $\text{dim}V=\text{dim}W$, and $T\in \mathcal{L}(V,W)$. Then: $T\text{is invertible}⟺T\text{is injective}⟺T\text{is surjective}$

Proof:

1. as we know that T is invertible iff it is bijective, we just need to show the remaining two imply each other
2. Assume $T$ is injective. Then we have the following:
   1. $\text{dim }V=\text{dim null }T+\text{dim range }T$
   2. We can then write this in terms of the range, which we have to show is equivalent to $W$
   3. $\text{dim range }T=\text{dim }V-\text{dim null }T=\text{dim }V=\text{dim }W$
3. Assume $T$ is surjective
   1. Then we need to prove that $\text{null }T$ is {0}
   2. $\text{dim null }T=\text{dim }V-\text{dim range }T=\text{dim }V-\text{dim }W=\text{dim }V-\text{dim }V=0$

Example Question

Prove that there exists a polynomial $p$ such that $((x^2+5x+7)p{)}^{\prime \prime }=q$ #todo if you want to later, show that its injective, there is a max degree so the dimentions of the two map are equal,

Another same dimention proof

Suppose $V$ and $W$ are finite dimensional vector spaces of the same dimension, $S\in \mathcal{L}(W,V)$ and $T\in \mathcal{L}(V,W)$, then $ST=I⟺TS=I$.

Isomorphism definition

1. An isomorphism is an invertible linear map (which is a bijective linear map)
2. Two vectors are isomorphic if there is an isomorphism from one vector space to the other one We use isomorphism to emphasize that two vector spaces are essentially the same.

Isomorphism condition

Two dimensional vector spaces over F are isomorphic $⟺$ they have the same dimension. #todo do this proof before you sleep, utilize the fundamental theorem of linear maps. Also define a map from the basis to basis.

By the condition above, you can note that any finite dimensional vector space is isomorphic to $F^n$, ${\mathcal{P}}_n(F)$ is isomorphic to $F^{(n+1)}$.

Additionally, lets say $\text{dim }V=n$ and $\text{dim }W=m$. Then, for some $T\in \mathcal{L}(V,W)$ we can show that $\mathcal{L}(V,W)$ is isomorphic to ${\mathbb{F}}^{n,m}$.

$\mathcal{L}(V,W)$ is isomorphic to ${\mathbb{F}}^{m,n}$

Suppose $v_1,....,v_n$ is a basis of $V$ and $w_1,...,w_m$ is a basis of $W$. Then $\mathcal{M}$ is an isomorphism between $\mathcal{L}(V,W)$ and ${\mathbb{F}}^{n,m}$ #todo prove

Dim of $\mathcal{L}(V,W)$

From previous proofs, $\text{dim }\mathcal{L}(V,W)=(\text{dim }V)(\text{dim }W)$

Matrix of a vector

Just the scalers of thje linear combination