# Null space, Range, Fundamental theorem of linear maps

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 $T:V\to W$
• $T(au+bv)=aT(u)+bT(v)$
• Matrix of linear map with respect to bases for $V$ and $W$
• Basis for $V$: $\{v_1,\ldots,v_n\}$
• Column $j$: coordinates of $T(v_j)$ with respect to basis of $W$
• We will start studying some important aspects of linear maps in this lecture

## Null space

$T:V\to W$ is a linear map. The null space of $T$, denoted null $T$, is defined as $\text{null } T=\{v\in V:Tv=0\}.$ null $T$: subset of vectors that get mapped to 0 by $T$

Examples

• $T=0$, null $T=V$
• $T=1$, null $T=\{0\}$
• $D$: Differentiation of polynomials
• null $D$ = constant polynomials
• Multiplication of polynomials by $x^2$
• null space $= \{0\}$
• $T(x,y)=x+2y$
• null $T=\{(x,y):x=-2y\}=\text{span}\{(-2,1)\}$
• $T(x,y,z)=(x,x+y)$
• null $T=\{(0,0,z):z\in\mathbb{R}\}$, $z$ axis

## Null space and injectivity

$T:V\to W$ is a linear map. null $T$ is a subspace of $V$.

Proof: If $u,v\in\text{null }T$, $au+bv\in\text{null }T$

Corollary: A linear map $T$ always maps $0$ to $0$.

A map $T:V\to W$ is said to be injective if $Tu=Tv$ implies $u=v$.

Examples: identity, multiplication by $x^2$ are injective; zero, differentiation are not injective

A linear map $T:V\to W$ is injective iff null $T=\{0\}$.

• $T$: injective
• if $v\in\text{null }T$, $Tv=0=T0$. So, $v=0$.
• null $T=\{0\}$
• $Tu=Tv$ implies $T(u-v)=0$. So, $u-v\in\text{null }T$ or $u-v=0$.

## Range

$T:V\to W$ is a map. The range of $T$, denoted range $T$, is defined as $\text{range } T=\{Tv:v\in V\}.$ range $T$: set of outputs of $T$

Examples

• $T=0$, range $T=\{0\}$
• $T=1$, range $T=V$
• $D$: Differentiation of polynomials
• range $D$ = all polynomials
• Multiplication of polynomials by $x^2$
• range = polynomials with constant zero and coefficient of $x$ zero
• $T(x,y)=x+2y$
• range $T=\mathbb{R}$
• $T(x,y,z)=(x,x+y)$
• range $T=\mathbb{R}^2$, $x{-}y$ plane

## Range and surjectivity

$T:V\to W$ is a linear map. range $T$ is a subspace of $W$.

Proof

• If $w_1,w_2\in\text{range }T$, we have $Tv_1=w_1$ and $Tv_2=w_2$

• $aw_1+bw_2=T(av_1+bv_2)\in\text{range }T$

A map $T:V\to W$ is said to be surjective if range $T=W$.

Examples

• identity, differentiation are surjective
• zero, multiplication by $x^2$ are not surjective
• $T(x,y)=x+2y$, $T(x,y,z)=(x,x+y)$ are surjective

## Fundamental theorem of linear maps

Suppose $V$ is finite-dimensional and $T:V\to W$ is a linear map. Then, range $T$ is finite-dimensional and $\text{dim } V = \text{dim null } T + \text{dim range } T$

Proof sketch

• $\{u_1,\ldots,u_k\}$: basis for null $T$

• $\{u_1,\ldots,u_k,v_1,\ldots,v_l\}$: extension of above basis to basis of $V$

• Show $\{Tv_1,\ldots,Tv_l\}$ is a basis for range $T$

## How do all linear maps work?

$T:V\to W$ is a linear map, $V$: finite-dimensional

• null $T$: mapped to $0$

• $\{u_1,\ldots,u_k\}$: basis for null $T$

• $\{u_1,\ldots,u_k,v_1,\ldots,v_l\}$: extension

• Vectors of the form: $av_1+$ vector from null $T$

• mapped to $a\;Tv_1$
• Vectors of the form: $bv_2+$ vector from null $T$

• mapped to $b\;Tv_2$
• and so on…

• Each mapping above is to linearly independent vectors