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Unformatted text preview: Dr. Doom Lin Alg MATH185O Section 1.2 Page 1 of 8 Chapter 1  Linear Equations and Matrices
Section 1.2  Gaussian Elimination Deﬁnition: A matrix that has the following 4 properties is said to be in
reduced rowechelon form: 1. If a row does not consist entirely of zeros, then the ﬁrst nonzero number
in the row is a 1. We call this a leading 1. 2. If there are any rows consisting entirely of zeros, then they are grouped
together at the bottom of the matrix. 3. In any two successive rows that contain leading 1’s, the leading 1 in
the lower row occurs farther to the right than the leading 1 in the higher row. 4. Each column that contains a leading 1 contains zeros everywhere else. If a matrix satisﬁes the ﬁrst 3 properties listed above, then we say that the
matrix is in rowechelon form. Note: A matrix in reduced rowechelon form is necessarily in rowechelon
form, but not conversely. Dr. Doom Lin Alg MATH185O Section 1.2 Page 2 of 8 Examples  Rowechelon and reduced rowechelon: @0 2 0 1 J}
0 3 —2 0 0 —1 2 1
$360308 000 0G 00012
00000 000 00 001—10
Tuis'm in ﬁEJ‘. 1w}. ‘6 ‘13 EXILE parka (FAG; cwérrfoﬁ 3) cam 32' (1.2.6.1? The reason why we’re interested in these forms is that the systems associated
with them have solutions that are really easy to ﬁnd.
Examples  Solutions of Linear Systems: A x1 x3 x. x; x,
2 1 —1 4 1 0 0 2
3 _2 5 9 0 1 0 1
4 —1 —1 6 0 0 1 1
1 —6 1 —3 0 0 0 0
C ‘1 C T REM) THE ERSV To ELL THE sum 0:: WWW
00 Mb‘ '
Sowrl'oo , Mr (115 srsﬁn \s
hcxumu/ The smc x\ = ’2.
2,: 4 x315 ‘ Dr. Doom Lin Alg MATH185O Section 1.2 Page 3 of 8 Recall the Elementary Row Operations:
1. Multiply a row by a non0 constant
2. Interchange two rows
3. Add a multiple of one row to another. Know these in your sleep! We use them in the following algorithm that gives us the (reduced) row
echelon form of the matrix we start with: 1. Locate the ﬁrst non—zero column (usually the ﬁrst). 2. In that column locate a leading 1, or create a leading 1 using an ele
mentary row operation. 3. Move the leading 1 to the top of that column, by switching 2 rows.
4. Use the leading 1 to create 0’s underneath it, using row operations.
5. Move on to the next column and repeat the process (steps 15) work ing now in the rows strictly below the one containing the leading 1 used in
the previous loop. The algorithm has a ﬁnite number of steps, and eventually we get the row—
echelon form; after which we use back substitution to solve for the variables. Gaussian elimination refers to the algorithm above that ends with a row
echelon form. Dr. Doom Lin Alg MATH185O Section 1.2 Page 4 of 8 Examples — Gaussian Elimination: w —— 3a: — 2y —— = 1
2w —— 4:1: — 2y — = 5
a: — 3y —— 6,2 = 9 map 1—41: ﬁFTEL um QANLE F00, GAUSS 5'09. on.) 3
\
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q
\ '5 2 t l
o ~7. 7.. ’3 3
o 3 —3 L ‘3
\
3
3 3' 4";
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_?~"_V2.; ° ‘ q 2' 3 2"” i a \ —\ o 45 7.3 LKEF
o —'L '2. —3 '3 a 0 o _l ‘\ v
w x y ”IR£0 (spa—vsuwma \6 “5‘“
Z
1 3 —'L l l
O \ \ 1. 3
O 0 z=q
23 [134—227. 0 o ‘\
x — +2.2. —?:
LET at ‘3'
Weaken. =2 x=s+.3—1¢.=3+£vl% u) ‘V‘X’Zlai' 2. = A
’3 UJ= \‘Ex+23Z.
= p3£k4©+1kvci 50— So 6606M. sourn‘on ‘ns uoc 37—71:. Dr. Doom Lin Alg MATH1850 Section 1.2 Page 5 of 8 Examples — GaussJordan Elimination:
This algorithm ends when we obtain the reduced rowechelon form 
i.e. we need to continue the previous algorithm and create zeros above the leading 1’s as well. \ \ O Pﬂé’ LL 2. 1, l 3 o l \ \ 0 2—2.“ 21—214 0 \ ~\
Q3‘. 2.5—3R\ 0 ‘3 l \ l O
l —\
_—’—’ o (I \ 23'. [23“ '2) O O
\ \ °
0 O \
l O O 1‘. R“ 21 o O ‘ 2:61  3:62 — $3 =
$1  $2 =
3331 + 333 =
'3
O
u
0
"3
ll
0
’3
l\
O
3
Z
O
3
\
O
A:
'l
H >513“
Lf xlcﬂ
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0
11 Dr. Doom Lin Alg MATH185O Section 1.2 Page 6 of 8 More examples of the tail end of the algorithm. Suppose we row reduced
the augmented matrix corresponding to some linear system and obtained: 0 0 2 0 0 1
0 0 0 Q 1 0 Yq'tX(=O=JXq=Xg so
0 0 0 0 0 0
x+lx3=x :9 )c‘2 \'2.x
m w 54::
x3=k
X55“ Some terminology: a leading variable is a variable that has a leading 1
in its column and a free variable is a variable that has no leading 1 in its
column Dr. Doom Lin Alg MATH185O Section 1.2 Page 7 of 8 Deﬁnition: A linear system is said to be homogeneous if all the con
stants are equal to 0, i.e. if it is of the form: a11$1  (“2:62  . . .  @1713?” = 0
(@1361  (@2362  . . .  @2713?” = 0
amlgcl __ am2$2 __    __ amngcn = 0 The trivial solution refers to the solution that has all variables equal to
zero; all homogeneous systems have (at the very least) the trivial solution.
Nontrivial solutions refers to solutions other than the trivial one. Some Properties of homogeneous linear systems: 0 All homogeneous systems are consistent (as they all have the trivial so
lution, at least). 0 For any homogeneous system, either the trivial solution is unique, or
there are inﬁnitely many solutions. Theorem: A homogeneous system with fewer equations than unknowns
has inﬁnitely many solutions. Examples  Homogeneous systems: II
o 2:61 — 3362 + $3
$1 + $2 — 2363 II
o Tms Wash was wMw soLuxws' Dr. Doom Lin Alg MATH185O Section 1.2 Page 8 of 8 Example: determine the values of k for which the following system has (i)
no solution, (ii) exactly one solution, (iii) inﬁnitely many solutions: M
as
I Pr [\D
<2
I  k+4 H
:
E
H
CO Q\<—:RL Ia \ IL 3
Z k1 \u—H \ ‘4 3
o kt—LV. La—L 21. mg 1. s b .
Ip \41u_+o wueu 9‘1: 21+h at) (.wss us A 2” Lemma am \ \L 3
‘—" i ] \LLk—L)
u \—
o 1 V
. 7.
So me am EXACTLV one Sow'run» “men \a—quﬁo
431,. k—éoﬁ, 515E (1Q \EZk =0 5.4. \z.=o on. ‘43).) ‘w lc=0 1160 (2.1 (Laos £0 a 4,3 J so stren MS
no $6LOTLBD. 1? b2 Tut.» Q.,_ tFAoS [a a 03,50 game. so an — mow? sot—UTE“ ~ ...
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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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