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# 1.2 - Dr Doom Lin Alg MATH185O Section 1.2 Page 1 of 8...

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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 row-echelon 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 row-echelon form. Note: A matrix in reduced row-echelon form is necessarily in row-echelon form, but not conversely. Dr. Doom Lin Alg MATH185O Section 1.2 Page 2 of 8 Examples - Row-echelon and reduced row-echelon: @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 ﬁ-EJ‘. 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 non-0 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 1-5) 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 \ S' q \ '5 -2 t l o ~7. 7.. ’3 3 o 3 —3 L ‘3 \ 3 3 3' 4"; “\$me 0 a | a | 5 "Z. \ l ‘ _?~"_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+2|3-Z. = 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 — Gauss-Jordan Elimination: This algorithm ends when we obtain the reduced row-echelon 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 " So X3=~l —3 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. Non-trivial 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 w-Mw soLux-ws' 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 \41-u_+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 \E-Zk =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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