NoteNextra-origin/pages/Math429/Math429_L12.md

Lecture 12

Chapter III Linear maps

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

Matrices 3C

Proposition 3.51

Let C be an 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}

Column-Row Factorization and Rank

Definition 3.52

Let A be an m \times n matrix, then

• The column rank of A is the dimension of the span of the columns in \mathbb{F}^{m,1}.
• The row range of A is the dimension of the span of the row in \mathbb{F}^{1,n}.

Transpose: A^t=A^T refers to swapping rows and columns

Theorem 3.56 (Column-Row Factorization)

Let A be an m\times j matrix with column rank c. Then there exists an m\times c matrix C and c\times x matrix R such that A=CR

Proof:

Let V=Span\{A_{\cdot,1},...,A_{\cdot,n}\}, let C_{\cdot, 1},...,C_{\cdot, c} be a basis of V. Since these forms a basis, there exists R_{j,k} such that A_{i,j}=\sum_{j=1}^c C_{i,j}R_{j,k}, so A_{\cdot,j}=\sum_{j=1}^c C_{\cdot,j}R_{j,k}. This implies that A=CR by construction C is m\times c, R is c\times n.

Example:

A=\begin{pmatrix}
    1&4&2\\
    2&5&8\\
    3&6&4
\end{pmatrix}=\begin{pmatrix}
    1&4\\
    2&5\\
    3&6\\
\end{pmatrix}\begin{pmatrix}
    1&0&-1\\
    0&1&2\\
\end{pmatrix},rank\ A=4

Definition 3.58 Rank

The rank of a matrix A is the column rank of A denoted rank\ A.

Theorem 3.57

Given a matrix A the column rank equals the row rank.

Proof:

Note that by Theorem 3.56, if A is m\times n and has column rank c. A=CR for some C is a m\times c matrix, R is a c\times n matrices, ut the rows of CR are a linear combination of the rows of R, and row rank of R\leq C. So row rank A\leq column rank of A.

Taking a transpose of matrix, then row rank of A^T (column rank of A) \leq column rank of A^T (row rank A).

So column rank is equal to row rank.

Invertibility and Isomorphisms 3D

Invertible Linear Maps

Definition 3.59

A linear map T\in\mathscr{L}(V,W) is invertible if there exists S\in \mathscr{L}(W,V) such that ST=I_V and TS=I_W. Such a S is called an inverse of T.

Note: ST=I_V and TS=I_W must both be true for inverse map.

Lemma 3.60

Every linear map has an unique inverse.

Proof: Exercise and answer in the book.

Notation: T^{-1} is the inverse of T

Theorem 3.63

A linear map T:V\to W invertible if and only if its injective and surjective.

Proof:

\Rightarrow

null(T)=\{0\} since T(v)=0\implies (T^{-1}))(T(v))=0\implies range (T)=W let w\in W then T(T^{-1}(w))=w,w\in range (T)

\Leftarrow

Find S:W\to V a function such that T(S(v))=v by letting S(v) be the unique vector in v such that T(S(v))=v. Goal: Show S:W\to V is linear

ST(S(w_1)+S(w_2))=S(w_1)+S(w_2)\\
S(T(S(w_1)))+T(S(w_2))=S(w_1+w_2)