Lecture 11

Chapter III Linear maps

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

Matrices 3C

Definition 3.31

Suppose $T\in \mathscr{L}(V,W)$, $v_1,...,v_n$ a basis for $V$ $w_1,...,w_m$ a basis for $W$. Then $M(T)=M(T,(v_1,...,v_n),(w_1,...,w_m))$ is given by $M(T)=A$n where

Example:

• $T:\mathbb{F}^2\to \mathbb{F}^3$

  $T(x,y)=(x+3y,2x+5y,7x+9y)$

  $M(T)=\begin{pmatrix} 1&3\\ 2&5\\ 7&9 \end{pmatrix}$

• Let $D:\mathscr{P}_3(\mathbb{F})\to \mathscr{P}_2(\mathbb{F})$ be differentiation

  $M(T)=\begin{pmatrix} 0&1&0&0\\ 0&0&2&0\\ 0&0&0&3\\ \end{pmatrix}$

Lemma 3.35

$S,T\in \mathscr{L}(V,W)$, $M(S+T)=M(S)+M(T)$

Lemma 3.38

$\forall \lambda\in \mathbb{F},T\in \mathscr{L}(V,W)$, $M(\lambda T)=\lambda M(T)$

$M:\mathscr{L}(V,W)\to \mathbb{F}^{n,m}$ is a linear map

Matrix multiplication

Definition 3.41

Theorem 3.42

$T\in \mathscr{L}(U,V), S\in\mathscr{L}(V,W)$ then $M(S,T)=M(S)M(T)$ ($dim (U)=p, dim(V)=n, dim(W)=m$)

Proof:

Let $w_1,...,v_n$ be a basis for $V$, $w_1,..,w_m$ be a basis for $W$ $u_1,..,u_p$ be a basis of $U$.

Let $A=M(S),B=M(T)$

Compute $M(ST)$ by Definition 3.31

Notation 3.44

Suppose $A$ is an $m\times n$ matrix

then

1. $A_{j,\cdot}$ denotes the $1\times n$ matrix at the $j$th column.
2. $A_{\cdot,k}$ denotes the $m\times 1$ matrix at the $k$th column.

Proposition 3.46

Suppose $A$ is a $m\times n$ matrix and $B$ is a $n\times p$ matrix, then

Proof:

$(AB)_{j,k}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$

$(A_{j,\cdot})\cdot (B_{\cdot,k})=(A_{j,\cdot})_{1,1}(B_{\cdot,k})_{1,1}+...+(A_{j,\cdot})_{1,n}(B_{\cdot,k})_{n,1}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$

Proposition 3.48

Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then

Proposition 3.56

Let $A$ is an $m\times n$ $b=\begin{pmatrix} b_1\\...\\b_n \end{pmatrix}$ a $x\times 1$ matrix. Then $Ab=b_1A_{\cdot,1}+...+b_nA_{\cdot,n}$

i.e. $Ab$ is a linear combination of the columns of $A$

Proposition 3.51

Let $C$ be a $m\times c$ matrix and $R$ be a $c\times n$ matrix, then

1. column $k$ of $CR$ is a linear combination of the columns of $C$ with coefficients given by $R_{\cdot,k}$

   putting the propositions together...

2. row $j$ of $CR$ is a linear combination of the rows of $R$ with coefficients given by $C_{j,\cdot}$