ST259 Lecture Notes - If And Only If

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20 Mar 2019

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4lDeterminantsoforderI

DefI

Aqbd is a22matrix withentriesfrom afield Fthen we define the

determinant of Adenoted det Aor lat tobethescalar adibc

Tha4.1

Thefunction Det Marz FFis alinearfunction of each row of a22matrix

whentheother row is heldfixedThat is if urand ware in FZand Kis ascalar

then

detfutwk detgkdetfw

and

det www detfwu tkdetfw

ThM4.2 Let A

22FThenthedeterminant of Ais nonzero iff Ais invertible Moreover

if Ais invertible then

AdefeatHa Aa

The Area of aParrellelogram

Bythe angle betweentwovectors in 1132 we mean the angle with measure 0CO

EOEt

that is formedbythevectors havingthe same magnitude and direction as thegiven

vectors but emanatingfromthe origin

If DuVis an ordered basis for 1132 we define theorientation of Bto be the

real number

det

Idet I

Clearly

Noticethat

0el Iand Ofer I

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