4.5.10 · D1Row echelon form and reduced row echelon form
Linear equations ka ek system basically variables ka ek ulajha hua jaal hai, aur poora topic ek hi trick pe tika hai: sirf reversible row moves use karo taaki us jaale ko ek staircase shape mein laaya ja sake, jisse har row ek ek variable reveal kare. Baaki sab — pivots, REF, RREF, free variables, inconsistency — bas us staircase ko describe karne aur usse answers padhne ki vocabulary hai.
Is page pe assume kiya gaya hai ki aapne parent note ki notation mein se kuch bhi nahi dekha. Hum har symbol neeche se upar build karenge, ek aisi order mein jahan har idea sirf usse pehle waale ideas pe depend karta hai.
Matrices se pehle, staircases se pehle, hoti hai linear equation .
Definition Variable aur coefficientEk variable ek unknown number hota hai jo hum find karna chahte hain — kisi letter jaise x y z coefficient ek known number hota hai jo variable ko multiply karta hai, jaise 3 x 3 Picture: variable ko ek khaali dabbe □
Ek equation jaise
x + 2 y + z = 6 humne jo bhi numbers dabbon mein daale, left side ka total right side ke barabar hona chahiye. Word linear ka matlab hai ki har dabba akela aata hai (power 1 x 2 x y x
Intuition "Linear" = "seedha" kyun?Un sabhi ( x , y ) x + 2 y = 6 seedhi line hoti hai — koi mod nahi. Yahi flatness ek aisi wajah hai ki poora subject simple rehta hai: seedhi cheezein predictable tareekon se intersect karti hain.
Definition System of linear equationsEk system linear equations ka ek aisa set hota hai jo sab ek saath hold hone chahiye. Ek solution un numbers ka ek aisa choice hai jo har line ka promise poora kare.
⎩ ⎨ ⎧ x + 2 y + z = 6 2 x + 5 y + 3 z = 15 x + 3 y + 3 z = 11
Curly brace { ka matlab hai "ye sab ek saath". Geometrically har equation space mein ek flat sheet hai; ek solution ek aisa point hai jo sab sheets pe ek saath baithta ho — ek intersection. Dekho Solving Systems of Linear Equations .
Intuition Hume ek nayi tool kyun chahiyeTeen equations, teen dabbe, har jagah tangled numbers. Hand-substitution se solve karna jaldi messy ho jaata hai. Hum ek mechanical, mistake-proof procedure chahte hain — aur uske liye pehle hum letters hata dete hain aur sirf numbers rakhte hain. Yahi matrix hai.
Ek matrix numbers ki ek rectangular grid hoti hai jisme rows (horizontal) aur columns (vertical) hote hain. Hum ise bade brackets [ ] Picture: ek spreadsheet. Row = =
Upar waala system lo. Coefficients ko column ke hisaab se line up karo — column 1 mein saare x y z
1 2 1 2 5 3 1 3 3
Kuch bhi throw away nahi kiya gaya sirf box-names ke alawa , aur hum unke order pe agree kar chuke hain (x, y, z). Numbers hain hi ab equations.
R i i R i i R 1 R 2 i kaun si row.
Equations mein = 6 , 15 , 11
Definition Augmented matrixEk augmented matrix coefficient grid aur constants column ko ek saath jodti hai, bar ∣
1 2 1 2 5 3 1 3 3 6 15 11
Common mistake Bar sirf "decoration" nahi haiYe harmless kyun lagta hai: ye kisi bhi doosre column jaisa dikhta hai. Fix: bar ke baad waale column mein constants hote hain, kisi variable ke coefficients nahi. Agar koi "pivot" baad mein wahan land kare to iska matlab contradiction hai (Section 8) — isliye bar change karta hai ki aap final grid ko kaise padhte hain.
Ab hum grid ko simplify karna chahte hain bina ye change kiye ki kaun se box-values use solve karte hain. Sirf teen moves allowed hain, aur har ek is liye choose kiya gaya hai kyunki wo reversible hai (aap use undo kar sakte hain), yahi exact wajah hai ki ye koi solution lose ya invent nahi kar sakta.
c = 0 0 0 = 0 mita diya , aur aap kabhi 0
"Ek row ka multiple add karna" safe kyun hai: ek sachi equation ke dono sides mein barabar amount add karna use sach rakhta hai, aur aap us amount ko subtract karke reverse kar sakte hain. In teeno moves mein se har ek elementary matrix se left-multiply karne ke corresponding hai — lekin abhi aapko wo nahi chahiye.
Ab show ka star.
Definition Pivot (leading entry)Ek non-zero row mein, pivot pehla non-zero number hota hai baaye se daayein padhne par. Jis column mein wo baithe, wo pivot column hai.
Picture: ek row ko left se scan karo; pivot pehla tile hai jo blank (0 0
Intuition Pivot kyun matter karta haiEk pivot us variable ko mark karta hai jisko ye equation "charge mein" rakhegi. Agar hum pivot ke neeche har entry ko zero bana sakein, to wo variable saari lower equations se gayab ho jaata hai — use isolate kar diya gaya upar ki taraf. In isolations ko stack karo aur aapko ek staircase milti hai jahan har step ek variable ka malik hota hai.
Ek zero row (sabhi 0 nahi hota — wo koi promise nahi karta (0 = 0
Condition 2 "neeche-aur-daayein" staircase hai; condition 3 wahi hai jo variables ko isolate karti hai. REF tak pahunchna exactly Gaussian Elimination hai.
Intuition Staircase kaisi dikhti haiPivots ko trace karte hue ek line kheencho: wo sirf kabhi daayein aur neeche step karti hai, kabhi wapas baayein nahi, kabhi upar nahi. Us line ke neeche: sab zeros. Uske upar aur uspe: jo kuch bacha. Sabse neeche pivot row sabse choti hoti hai — aksar sirf ek variable = ek number — wahi seed hoti hai jahan se aap upar ki taraf back-substitute karte hain.
Definition Reduced Row Echelon Form (RREF)RREF, REF plus hai:
4. har pivot 1 leading 1 );
5. har pivot apne column mein single non-zero entry ho (upar aur neeche zeros).
Is tak pahunchna Gauss-Jordan Elimination hai. Faida: har pivot row directly padhti hai "variable = unique hoti hai, jabki REF nahi hoti (kai staircases, ek dusted form).
REF/RREF mein aane ke baad pivots count karo. Pivots ki sankhya rank hai.
Ek solution: har variable column mein ek pivot, constants column mein koi nahi.Koi solution nahi (inconsistent): constants column mein ek pivot — ek row 0 = nonzero Solving Systems of Linear Equations .Infinitely many: kisi variable column mein koi pivot nahi → wo variable ek free variable hai, kuch bhi ho sakta hai, aur baaki sab uspe depend karta hai.
Intuition Kam pivots ⇒ freedom kyunHar pivot ek variable ko pin karta hai. Agar aapke paas pivot-columns se zyada variable-columns hain, to kuch columns un-pinned hain — woh variables float karte hain, solutions ki poori ek line ya plane trace karte hue. Pivot columns, is ke contrast mein, linearly independent directions hain.
Variables and coefficients
Linear equation = straight line
Elementary row operations
Read off solutions rank freedom
"Linear" word har variable ko equation mein kaisi force karta hai? Wo akela aata hai, sirf pehli power pe — koi squares nahi, koi products nahi, koi roots nahi — isliye graph ek straight line hoti hai.
Coefficient aur variable mein kya fark hai? Coefficient ek known number hota hai jo variable ko multiply karta hai; variable wo unknown box hai jise hum solve karte hain.
R i Matrix ki row number i
Augmented matrix ki last column ko bar ke baad kyun draw kiya jaata hai? Usme constants (right-hand sides) hote hain, kisi variable ke coefficients nahi — alag tarike se padhte hain.
Teen elementary row operations ke naam batao. Do rows swap karo; ek row ko ek non-zero constant se scale karo; ek row ka multiple doosri row mein add karo.
Scale constant non-zero kyun hona chahiye? 0
Teeno operations solution set ko preserve kyun karti hain? Har ek reversible hai, isliye koi solution gain ya lose nahi ho sakta.
Pivot kya hota hai? Ek non-zero row mein pehla non-zero entry (baaye se daayein padhte hue).
Kya zero row mein pivot hota hai? Nahi — ek all-zero row koi promise nahi karti aur koi pivot contribute nahi karti.
REF mein pivots kaisi shape trace karte hain? Ek staircase jo sirf daayein aur neeche step karti hai, har pivot ke neeche zeros ke saath.
REF ke upar RREF jo do extra conditions add karta hai? Har pivot 1
Kaun sa form unique hota hai — REF ya RREF? Sirf RREF ek diye gaye matrix ke liye unique hoti hai.
Constants column mein pivot ka matlab kya hai? System inconsistent hai — ek row 0 = nonzero
Free variable kya hota hai? Ek aisa variable jiske column mein koi pivot nahi; wo koi bhi value le sakta hai.
