Lecture 10

Chapter III Linear maps

Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)

Vector Space of Linear Maps 3A

Review

Theorem 3.21 (The Fundamental Theorem of Linear Maps, Rank-nullity Theorem)

Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V,W)$, then $range(T)$ is finite dimensional ($W$ don't need to be finite dimensional). and

Proof:

Let $u_1,...,u_m$ be a basis for $null(T)$, then we extend to a basis of $V$ given by $u_1,...,u_m,v_1,...,v_m$, we have $dim(V)=m+n$. Claim that $Tv_1,...,Tv_n$ forms a basis for $range (T)$. Need to show

• Linearly independent. (in Homework 3)

• These span $range(T)$.

  Let $w\in range(T)$ the there exists $v\in V$ such that $Tv=W$, $u_1,...,u_m,v_1,...,v_m$ are basis so $\exists a_1,...,a_m,b_1,...,b_n$ such that $v=a_1u_1+...+a_mu_m+b_1v_1+...+b_n v_n$. $Tv=a_1Tu_1+...+a_mTu_m+b_1Tu_1+...+b_nTv_n$.

  Since $u_k\in null(T)$, So $Tv_1,...,Tv_n$ spans range $T$ and so form a basis. Thus $range(T)$ is finite dimensional and $dim(range(T))=n$. So $dim(V)=dim(null (T))+dim(range(T))$

Theorem 3.22

Suppose $V,W$ are finite dimensional with $dim(V)>dim(W)$, then there are no injective maps from $V$ to $W$.

Theorem 3.24

Suppose $V,W$ are finite dimensional with $dim(V)<dim(W)$, then there are no surjective maps from $V$ to $W$.

ideas of Proof: relies on Theorem 3.21 $dim(null(T))>0$

Linear Maps and Linear Systems 3EX-1

Suppose we have a homogeneous linear system * with $m$ equation and $n$ variables.

which is equivalent to

also equivalent to

Solution to * is $null(T)$.

Proposition 3.26

A homogeneous linear system with more variables than equations has non-zero solutions.

Proof:

Using $T$ as above, note that since $n>m$, use Theorem 3.22, implies that $T$ cannot be injective. So, $null (T)$ contains a non-zero vector.

Proposition 3.28

An in-homogenous system with more equations than variables has no solutions for some choices of constants. ($A\vec{x}=\vec{b}$ for some $\vec{b}$ this has no solution)

Matrices 3A

Definition 3.29

For $m,n>0$ and $m\times n$ matrix $A$ is a rectangular array with elements of the $\mathbb{F}$ given by

Operations on matrices

Addition:

for $A+B$, $A,B$ need to be the same size

Scalar multiplication:

Definition 3.39

$\mathbb{F}^{m,n}$ is the set of $m$ by $n$ matrices.

Theorem 3.40

$\mathbb{F}^{m,n}$ is a vector space (over $\mathbb{F}$) with $dim(\mathbb{F}^{m,n})=m\times n$

Matrix multiplication 3EX-2

Let $A$ be a $m\times n$ matrix and $B$ be an $n\times s$ matrix

Claim:

This formula comes from multiplication of linear maps.

Definition 3.44

Linear maps to matrices, let $V$, $W$, $Tv_i$ written in terms of $w_i$.