Invertible maps, Isomorphism, Operators

Aug-Nov 2020

Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• Linear map $T:V\to W$ preserves linear combinations
• null $T=\{v\in V:Tv=0\}$, range $T=\{Tv:v\in V\}$
• Fundamental theorem of linear maps
• dim null $T$ + dim range $T$ = dim $V$
• $m\times n$ matrix A represents a linear map $T:F^n\to F^m$
• colspace(A) = range T, null(A) = null(T)
• Matrix-vector multiplication, matrix multiplication

Invertible linear maps

A linear map $T:V\to W$ is said to be invertible if there exists a linear map $S:W\to V$ (called inverse of $T$) such that $ST:V\to V$ is the identity map on $V$ and $TS:W\to W$ is the identity map on $W$.

Properties

• An invertible linear map has a unique inverse.

• Suppose $S_1,S_2$ are two inverses. $S_1=S_1I=S_1(TS_2)=(S_1T)S_2=IS_2=S_2$

• $T^{-1}$: denotes unique inverse when it exists

• $T$ is invertible iff it is injective and surjective.

• This is a classic general result about maps.

• See book for proof.

Examples

• Maps that are not injective are non-invertible

• Pick any $T$ with null $T\ne \{0\}$
• Maps that are not surjective are non-invertible

• Pick any $T$ with range $T\ne W$
• Polynomials: multiplication by $x^2$

• Injective, not surjective, non-invertible
• Polynomials: left shift

• Not injective, surjective, non-invertible

Isomorphism

An invertible linear map is called an isomorphism. Two vector spaces are called isomorphic if there is an isomorphism between them.

Two finite-dimensional vector spaces over the same field $F$ are isomorphic iff they have the same dimension.

• Proof
• $V,W$: isomorphic
• There exists an invertible $T:V\to W$
• dim null $T=0$, dim range $T=$ dim $W$
• Fundamental theorem: dim $V=$ dim $W$
• dim $V=$ dim $W$
• Define $T$ by mapping basis to basis
• $T$ is invertible

A finite-dimensional vector space $V$ is isomorphic to $F^n$, where $n=$ dim $V$.

Isomorphism of linear maps and matrices

dim $V=n$, dim $W=m$, $\mathcal{L}(V,W)$: vector space of linear maps, $F^{m,n}$: vector space of $m\times n$ matrices

$\mathcal{L}(V,W)$ and $F^{m,n}$ are isomorphic.

• Fix bases for $V$ and $W$

• Define map $\mathcal{M}:\mathcal{L}(V,W)\to F^{m,n}$ as follows.
• $T:V\to W$ mapped to matrix with respect to chosen bases.
• $\mathcal{M}$: linear, injective, surjective

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

Operators

A linear map from a vector space to itself is called an operator.

$\mathcal{L}(V)$: set of all operators on $V$.

• Operators are the most important linear maps.

• Invertible operators or invertible square matrices are an important class.

• When are operators injective, surjective and invertible?

• Polynomials: multiplication by $x^2$
• Injective, not surjective, non-invertible
• Polynomials: left shift
• Not injective, surjective, non-invertible
• Finite dimensions?

Invertible operators in finite dimensions

Let $T:V\to V$ be an operator and let $V$ be finite-dimensional. Then the following are equivalent: (a) $T$ is invertible; (b) $T$ is injective; (c) $T$ is surjective.

Proof

• a implies b: by definition

• b implies c: Fundamental theorem

• dim null $T=0$; so, dim $V=$ dim range $T$
• c implies a: Fundamental theorem

• dim range $T=$ dim $V$; so, dim null $T=0$

Invertibility of matrices

How to find if a matrix is invertible?

• Matrix has to be square

• $m>n$: not surjective

• $m<n$: not injective

• If square, the column rank has to be full

• Find dimension of column space

• Gaussian elimination