Gaussian elimination — forward elimination, back substitution: Foundations

4.5.9 · D1 · Maths › Linear Algebra (Full) › Gaussian elimination — forward elimination, back substitutio

Elimination karne se pehle aapko ek chhoti si alphabet mein fluent hona chahiye. Yeh page uske har letter ko kuch bhi nahin se build karta hai. Upar se neeche padho — har idea agla use karta hai.

1. Numbers ko ek grid mein rakhna — matrix

Picture yeh socho: ek spreadsheet. Har number ek fixed street address par rehta hai jo (konsi row, konsa column) se diya jata hai.

Figure — Gaussian elimination — forward elimination, back substitution

2. Ek single entry ko naam dena — subscript $a_{ik}$

To $a_{23}$ hai row 2, column 3 (Figure 1 mein red box wali entry). Ise " $a$ sub two three" padho, kabhi "twenty-three" mat kaho.

3. Diagonal aur uske neeche kya hai

Us line ke lower-left mein sab kuch "below the diagonal" hai. Forward elimination ka poora kaam yeh hai ki un saari below-diagonal entries ko zero bana de.

4. Bold letters — vector $x$ aur $b$

$x = x y z, b = 8 - 11 - 3$

Bold font ek promise hai: "yeh poori stack hai, ek number nahin." Plain $x$ ek number hai; bold $x$ saare unknowns ki list hai.

5. Grid ko stack se multiply karna — $A x$

Row $i$ ke liye: $a_{i 1} x_{1} + a_{i 2} x_{2} + \dots + a_{in} x_{n}$ . Woh sum exactly equation $i$ ka left-hand side hai. (Isliye columns ki sankhya $n$ , $x$ ki length ke barabar honi chahiye: har column entry ko $x$ mein ek partner chahiye.)

6. Unhe saath pack karna — augmented matrix $[A ∣ b]$

$2 - 3 - 2 1 - 1 1 - 1 22 8 - 11 - 3$

7. Teen legal moves — row operations

Arrow $\to$ ka matlab hai "ban jaata hai". $R_{i}$ "row $i$ poori line ke taur par" naam deta hai. Do-taraf wala arrow $\leftrightarrow$ ka matlab hai "jagah badalna."

8. Show ka star — pivot aur multiplier $m_{ik}$

Yahan fraction bar ordinary division hai. Hum us entry ko pivot se divide karte hain jo zero karni hai, kyunki woh ratio batata hai "row $i$ mein is column ki kitni pivot-rows hain."

9. Sum shorthand — symbol $\sum$

Jab forward elimination khatam ho jaati hai, $A$ upper-triangular matrix $U$ ban chuka hota hai, aur $b$ ek nayi right-hand column ban chuki hoti hai jise hum $c$ rename karte hain (kyunki har row operation ne ise bhi badla). Inke pieces ko hum yeh naam dete hain:

To $\sum_{j = 2}^{3} u_{1 j} x_{j} = u_{12} x_{2} + u_{13} x_{3}$ : bas ek compact tarika hai lamba " $+ + +$ " chain likhne ka jab hume pehle se nahi pata kitne terms honge.

10. Final staircase padhna — back substitution

Section 9 ke naamon ka use karke, jab $A$ staircase $U$ ban chuka ho nayi right side $c$ ke saath, bottom row mein ek single unknown hota hai, to tum use directly solve karo; phir upar chadhte jao, har row ek naya unknown deta hai.

$x_{n} = \frac{c _{n}}{u _{nn}}, x_{i} = \frac{1}{u _{ii}} (c_{i} - \sum_{j = i + 1}^{n} u_{ij} x_{j}) .$

Yahan $u_{ii}$ $U$ ki row $i$ mein diagonal (pivot) entry hai, $u_{ij}$ us row ke saath baaki entries hain, aur $c_{i}$ transformed right-hand number hai. $u_{ii}$ se divide karna final "akele unknown ke liye solve karo" step hai — isliye exactly $u_{ii}$ zero nahi hona chahiye.

Recall Symbol speed-check

$a_{31}$ kahan point karta hai? ::: Row 3, column 1. $[A ∣ b]$ mein bar kya separate karta hai? ::: Coefficients (baayein) ko right-hand-side values (daayein) se. Pivot nonzero kyun hona chahiye? ::: Kyunki hum ise multiplier $a_{ik} / a_{k k}$ mein divide karte hain. Agar pivot $a_{k k} = 0$ ho to kya karte ho? ::: Us column mein nonzero entry wali lower row swap karo; agar koi na ho, column skip karo (free variable). $c$ kya hai? ::: Right-hand column $b$ forward elimination ke baad transform hone par. $\sum_{j = i + 1}^{n}$ $j$ ko kya karne kehta hai? ::: $i + 1$ se $n$ tak chalo, har term jodते hue.

Foundations topic ko kaise feed karte hain

Equipment checklist

Daayein side cover karo; tum parent note ke liye ready ho sirf tab agar har ek ka jawab de sako.

Main entry $a_{ik}$ ko grid mein locate kar sakta hoon ::: row $i$ , column $k$ — pehle neeche, phir across.
Main jaanta hoon index letters ka kya matlab hai ::: $i$ = row ( $1 \dots m$ ), $j$ = column ( $1 \dots n$ ), $k$ = current pivot column.
Main jaanta hoon $x$ ki length $n$ kyun hai ::: $A$ ke $n$ columns hain, ek har unknown ke liye, to $x$ ko $n$ entries chahiye; $b$ ke $m$ entries hain, ek har row ke liye.
Main jaanta hoon "upper-triangular" kaisa dikhta hai ::: main diagonal ke neeche sab zeros; nonzeros diagonal par aur uske upar triangle banate hain.
Main $A x = b$ ko ordinary equations mein wapas badal sakta hoon ::: $A$ ki har row ko $x$ se multiply karo aur $b$ ki matching entry ke barabar set karo.
Main jaanta hoon augmented matrix kyun use karte hain ::: taaki right-hand side har row operation mein coefficients ke saath saath badle.
Main teen legal row operations state kar sakta hoon ::: swap, nonzero $c$ se scale, aur $R_{i} \to R_{i} - m R_{j}$ se replace.
Main multiplier bina yaad kiye derive kar sakta hoon ::: $a_{ik} - m_{ik} a_{k k} = 0$ set karo, solve karke $m_{ik} = a_{ik} / a_{k k}$ milta hai.
Main jaanta hoon zero pivot ke saath kya karna hai ::: us column mein nonzero entry wali lower row swap karo; agar koi na ho, column skip karo.
Main $u_{ij}$ aur $c_{i}$ ke naam jaanta hoon ::: finished upper-triangular $U$ ki entries aur transformed right-hand column $c$ ki.
Main $\sum$ expand kar sakta hoon ::: start value se end value tak $j$ ke liye har term likho aur jodo.

Connections

Solving Linear Systems — Consistency — final staircase tumhe kya batata hai no / one / infinite solutions ke baare mein.
Row Echelon Form / Reduced Row Echelon Form — woh tidy shapes jin par yeh foundations aim karti hain.
LU Decomposition — jahan multipliers $m_{ik}$ store aur reuse hote hain.
Pivoting and Numerical Stability — kyun "pivot nonzero hona chahiye" "pivot bada hona chahiye" mein grow karta hai.
Rank of a Matrix — yahan banaye pivots count karna.
Determinant — $U$ ke diagonal se padh lo jab staircase done ho.