Chapter 1
Linear Algebr a Revie w
The purpos e o f thi s chapte r i s t o revie w som e fundamenta l concept s
and result s fro m linea r algebr a an d matri x theor y tha t wil l be neede d
in th e mai n chapter s o f thi s book . A wid e variet y o f additiona l re -
sources i s availabl e fo r student s wh o wan t a mor e detaile d o r mor e
advanced treatmen t o f these topics . I n particular , w e refer th e reade r
to [4 ] and [45] . Th e exercise s a t th e en d o f thi s chapte r ar e intende d
to remin d th e reade r o f some o f the skill s learned i n thei r firs t semes -
ter o f linea r algebra . I f a studen t ca n wor k hi s o r he r wa y throug h
these exercises, with accurac y an d understanding , the n Chapte r 1 can
be skippe d excep t fo r gainin g familiarit y wit h ou r notation .
1.1. Vecto r Space s
We will consider vecto r space s over the real numbers R or the comple x
numbers C . W e will us e th e notatio n F whe n th e resul t bein g state d
holds fo r bot h R an d C . Element s o f F ar e calle d scalars.
Given a G C, w e will denot e th e comple x conjugat e o f a b y a
and th e modulu s o f a b y |a| .
Definition 1.1 . A vector space V is a nonempt y se t wit h tw o opera -
tions: additio n " + " an d multiplicatio n " " b y scalar s suc h tha t th e
following condition s ar e satisfie d fo r an y x,y,z G V an d an y a,/ ? i n
F.
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http://dx.doi.org/10.1090/stml/040/02
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