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MATH 3A (10)
Chapter 1

MATH 3A Chapter Notes - Chapter 1: Augmented Matrix, Linear Combination


Department
Mathematics
Course Code
MATH 3A
Professor
Wencai Liu
Chapter
1

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Ssl -4 Matrix Equation
Def.AI =[ai ,ai ..-Jen]-1¥!)-Xiaitxzazt ---txnan
Ex .[to I]-1431=4%+3133+7 II ]=[I]Matrix xvector -
-vector
II. Item III:
".tt Amin rn rn
Xi -12×2-73=4 (Linear System)x,[to ]+Xz[}]tX3[}]=[4)(vector form)
-5×2+3×3=1
[to -3-31](Yy
;] =[Y](matrix form )[alias ,as ----an
,
b](Augmented matrix )
matrix equating vector equation a
Theorem .
.Aman ,columns at ,at ---air ,IERM ,AI =D has same solution of Xia that ..-txnau -15
Note :AZ =D has asolution if Iis alinear combination of iolumns of A
nun murmur
Exz .Let A=[13%14]and b-[¥!].Is equation AI=I for all possible b?
'
÷÷ !:D .it:31#::s:.hi..it:IIoH::::............k*n- ±
If +bi -BI to ,so .no solution 󲍻AT =D is not consistent for all 15
Theorem :Amin ,the following statements are equalident
.
sina.ci#:iinae:::n:::io:::io:nn...f "÷÷i÷÷÷÷¥÷i÷
c)The columns of Aspan Rm
d)[Ab]has asolution for all bERM
Exz .A-
-I}Y],
a- IT ],v=[I]
25=19],
I+E- [I]
Ait :III -
-I:If :3 .Aoi 's III 's f- Eti 's f- It:3
Anita -[3,55141--13174]=17]-Art At
Aizu )=I}I]f- [Iz]=2AT
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