Math 307
Spring, 2005
Hentzel

Time: 10:00 to 10:40 MWF
Room: 205 Carver Hall
Instructor: Irvin Roy Hentzel
Office 432 Carver
Phone 515-294-8141
E-mail: hentzel@iastate.edu

http://orion.math.iastate.edu/hentzel/class.307.05


Text:  Linear Algebra and its Applications, Third Edition
       David C. Lay


Assignment:  Page 132 Problem 7, 8, 11, 12, 33 


Friday, February 4     2.3


Main Idea:   All for one and one for all. 
           

Key Words:     

invertible,               row equivalent,         stairstep ones, 

trivial solution,         linearly independent,   one-to-one, 

at least one solution,    spanning,               onto, 

left inverse,             right inverse,          transpose. 

Goal:  Learn all the stuff that comes with invertibility. 
------------------------------------------------------------

Theorem 8  The Invertible Matrix Theorem

Let A be a SQUARE NxN matrix.  

The following statements are equivalent.   

a.  A is an invertible matrix

b.  A is row equivalent to the nxn identity matrix

c.  A has n pivot positions

d.  The equation AX = 0 has only the trivial solution

e.  The columns of A form a linearly independent set.

f.  The linear transformation X ---> AX is one-to-one

g.  The equation AX = B has at least one solution for each B in R^n

h.  The columns of A span R^n
                                             n        n
i.  The linear transformation X --->AX maps R   ONTO R  

j.  There is an nxn matrix C such that CA = I.

k.  There is an nxn matrix D such that AD = I.

l.  A^T is an invertible matrix.

---------------------------------------------------------

It is true that the RCF(A) is unique.  But we have not

given a proof of that yet.  So now we will proceed

saying that

 "If one reduction of A to RCF gives I then  A is invertible."


A set of vectors X1 X2 ... Xn is called linearly dependent if

there exists a set of coefficients  c1, c2, ..., cn  not all zero

such that   
         
         c1 x1 + c2 X2 + ... cn Xn = 0  

The set of vectors is called linearly independent is no such

set of coefficients exists.

-------------------------------------------------------------
                                                               n
A set of vectors X1 X2 ... Xk is said to be a spanning set of R  

                        n
if for any vector B in R   there exists coefficients c1, c2, ..., cn

such that c  X  + c  X  + ... + c  X  = B.
           1  1    2  2          k  k
---------------------------------------------------------------

There are three basic conditions:

(1)  RCF(A) = I

     a.  A is an invertible matrix

     b.  A is row equivalent to the nxn identity matrix

     c.  A has n pivot positions

     j.  There is an nxn matrix C such that CA = I.

     k.  There is an nxn matrix D such that AD = I.

     l.  A^T is an invertible matrix.

(2)  The Columns of A are linearly independent.

     d.  The equation AX = 0 has only the trivial solution

     e.  The columns of A form a linearly independent set.

     f.  The linear transformation X ---> AX is one-to-one

                            n
(3)  The Columns of A span R  .
                                                                      n
     g.  The equation AX = B has at least one solution for each B in R  
                                n
     h.  The columns of A span R  
                                                  n        n
     i.  The linear transformation X --->AX maps R   ONTO R