Visual walkthrough — Pivot positions, free variables

4.5.11 · D2 · Maths › Linear Algebra (Full) › Pivot positions, free variables

Yahan sab kuch Pivot positions, free variables ko elaborate karta hai. Mechanics ke liye hum Row Reduction and Echelon Forms ka sahara lete hain aur aage Null Space and Solution Sets of Ax=0 aur Rank and the Rank-Nullity Theorem ki taraf point karte hain.

Step 1 — Ek linear system aslaan hota kya hai (boxes ki ek grid)

KYA HAI. Koi bhi symbol aane se pehle, raw object ko picture karo. $x_1,x_2,x_3$ variables mein linear equations ka ek system bas numbers ki ek table hai. Har row ek equation hai; har column un saare coefficients ko rakhta hai jo ek particular variable ko multiply karte hain.

KYUN. Hume "row" aur "column" shabdon ko ek picture se anchor karna hai pehle ki hum kabhi "pivot" bolein, kyunki ek pivot ek specific row aur ek specific column ke crossing par rehta hai. Agar ye dono words fuzzy hain, toh baad mein sab kuch fuzzy hai.

PICTURE. Figure mein, orange band ek row hai (ek equation). Teal band ek column hai (variable $x_2$ se attached sab kuch). Woh number jo wahan baithta hai jahan dono cross karte hain — woh single box — coefficient $a_{ij}$ hai: row $i$, column $j$.

Yahan abhi kuch bhi solve nahi hua hai. Humne bas parts ko naam diya.

Step 2 — Row reduction: grid ko staircase banana

KYA HAI. Ab hum grid ko staircase shape (echelon form) mein sweep karte hain. Row Reduction and Echelon Forms ke moves use karke — rows swap karo, ek row ko scale karo, ek row ka multiple doosri mein add karo — hum har row ka pehla nonzero number uske upar wale se strictly right mein bitha dete hain.

KYUN. Iska kya fayda? Kyunki staircase woh shape hai jismein har equation exactly ek variable reveal karti hai jise woh control karti hai — woh variable jo apne step ki tip par baithti hai. Woh tip hi is page ka pura point hai. Ek random grid chhupa leti hai ki kaun kis ko control karta hai; staircase ise obvious bana deta hai.

PICTURE. Plum "staircase edge" ko right ki taraf neeche utarte dekho. Har step ka corner us row ki leading entry hai — echelon form mein reduction ke baad uska pehla nonzero number.

$\underset{\text{first nonzero of row }i}{\underbrace{a_{ij}}}=\text{leading entry}\Rightarrow \text{column }j\text{ is a pivot column.}$

Step 3 — Har pivot exactly ek variable ko pin karta hai

KYA HAI. Staircase ki ek row lo. Uski leading entry kisi column $j$ mein baithti hai. Us row ko rearrange kiya ja sakta hai taaki yeh padhe "$x_j$ equals (doosre variables se bani cheez)". Toh column $j$ par pivot $x_j$ ko solve karta hai — woh us ek variable ko pin kar deta hai.

KYUN. Exactly ek kyun, kabhi do nahi? Kyunki "leading" ka matlab hai pehla. Us row mein leading entry ke left mein sab kuch pehle se zero hai (yahi staircase guarantee karta hai), isliye koi earlier variable is equation mein appear nahi karta. Leading variable hi woh akela hai jis ki zimmedari yeh equation lete hai; baaqi saare right-hand side par dhakele jaate hain.

PICTURE. Orange arrow pivot box se us variable ki taraf point karta hai jo woh capture karta hai. Uske left mein greyed-out zero boxes dekho — isliye koi earlier variable yeh equation cheen nahi sakti.

$\underset{\text{leading 1 in col }j}{\underbrace{1}}\cdot x_j+\underset{\text{later variables, moved right}}{\underbrace{c_{j+1}{x}_{j+1}+\cdots }}=d⟹x_j=d-(c_{j+1}{x}_{j+1}+\cdots )$

• $1$ leading entry hai (RREF mein pehle se $1$ tak scale ki hui).
• $x_j$ — woh pivot variable jis par yeh row ownership rakhti hai.
• Uske baad sab kuch — later variables, jinhe hum "known" maanke right mein dhakelte hain agar bata diye jaayein.

Step 4 — Bache hue columns free knobs ban jaate hain

KYA HAI. Kuch columns kabhi leading entry nahi paate — staircase unhe skip karte hue right mein chali jaati hai. Step 2 par wapas dekho: staircase ne ek column skip kiya. Koi row us variable ki charge mein nahi hai. Toh kuch bhi use pin nahi karta. Hum ise koi bhi number set kar sakte hain jaise chaahein.

KYUN. Ise "free" kyun bolte hain? Kyunki equations ise kabhi constrain nahi karti. Yeh ek knob hai: ise koi bhi value par ghumaao, aur pivot variables (Step 3) simply adjust ho jaate hain taaki har equation true rahe. Yahan ek parameter ka janam hota hai.

PICTURE. Teal column mein koi staircase corner nahi hai — dial icon use free knob ki tarah mark karta hai. Dial ghumaao aur orange pivot variables compensate karne ke liye slide karte hain.

Step 5 — Counting: columns exactly do piles mein split hoti hain

KYA HAI. Har column do kism mein se ek hoti hai, koi overlap nahi aur kuch bacha nahi: ya toh pivot hold karti hai ya nahi karti. Toh total column count $n$ saaf tarike se (pivot columns) + (non-pivot columns) mein split hoti hai.

KYUN. Yeh clean split kyun hai? Kyunki "pivot hai" ek column ke baare mein yes/no question hai — koi third option nahi hai. Ek column aadha-pivot nahi rakh sakti. Yahi exhaustiveness exactly hai jo hume subtract karne deti hai.

PICTURE. Saari $n$ columns line up mein. Orange = pivot columns (inki count $r$ hai). Teal = baaki. Dono colours poori strip ko bina gaps aur bina overlaps ke tile karte hain.

Yeh parent note ka headline result hai, ab assert nahi balki dekha gaya.

Step 6 — Free count ka geometric matlab kya hai

KYA HAI. Free knobs ki count solution set ki dimension hai (jab koi solution exist karta hai). Zero knobs → ek single point. Ek knob → ek line. Do knobs → ek plane.

KYUN. $k$ independent knobs ghumaana ek $k$-dimensional shape kyun sweep karta hai? Kyunki har free variable ek independent direction hai jis mein aap slide kar sakte ho, aur har independent slide ek dimension add karti hai — exactly Basis and Dimension ka idea. Ek slide ek line trace karti hai; ek doosri, independent slide line ko fan out karke plane bana deti hai.

PICTURE. Teen panels: $0$ knobs (ek dot), $1$ knob (ek line with arrow $t\mathbf{v}$), $2$ knobs (ek plane spanned by two arrows).

$\mathbf{x}=\underset{\text{one solution}}{\underbrace{\mathbf{p}}}+\underset{k\text{ free knobs}=k\text{ directions}}{\underbrace{t_1{\mathbf{v}}_1+\cdots +t_k{\mathbf{v}}_k}}$

${\mathbf{v}}_i$ woh directions hain jismein free variables push karti hain — null space ka ek basis jab $\mathbf{b}=0$.

Step 7 — Degenerate cases jinhe hum skip nahi kar sakte

KYA HAI. Formula $\text{free}=n-r$ ke boundary behaviours hain. Hum har ek check karte hain taaki koi reader wall se na takraaye.

KYUN. Ek rule jise aap extreme cases par push nahi kar sakte, woh ek aisa rule hai jis par aap trust nahi karte. Yahan saare edges hain.

PICTURE. Chaar mini-grids: (a) $r=n$ full rank, (b) zero rows present lekin full column rank, (c) equations se zyada variables, (d) ek inconsistent augmented column.

• (a) $r=n$ (koi free variables nahi). Har column mein pivot hai; $\text{free}=n-n=0$. At most ek solution — ek point (Step 6, left panel).
• (b) Zero rows appear hoti hain. Ek zero row ek free variable nahi banati. Free variables non-pivot columns se aati hain, kabhi empty rows se nahi. Figure mein, ek zero row neeche baithti hai phir bhi har column pivot hai — $\text{free}=0$. Yeh parent ki pehli steel-manned galti hai, draw ki gayi.
• (c) Wide matrix, $n>m$. Equations se zyada variables hone par, at most $m$ pivots exist karte hain, isliye $r\le m<n$, forcing $\text{free}=n-r\ge 1$. Hamesha at least ek free variable hogi — ek aisa system jisme unknowns equations se zyada hain woh sab kuch pin kabhi nahi kar sakta.
• (d) Inconsistent. Agar augmented column $[\mathbf{b}]$ mein khud pivot ho — ek row jo padhti hai $[0\cdots 0\mid c]$ jahan $c\ne 0$ — woh false statement $0=c$ hai. Solution set empty hai, aur free count kuch bhi describe nahi karta. Knobs count karne se pehle hamesha consistency check karo.

Ek-picture summary

Yeh final figure saatoon steps ko ek flow mein fold karta hai: raw grid → staircase → pivots variables pin karte hain → leftover columns free float karte hain → unhe $n-r$ ke roop mein count karo → solution ki dimension padho, ek consistency check se guard karke.

Recall checkpoints

Connections

• Pivot positions, free variables — parent topic
• Row Reduction and Echelon Forms
• Rank and the Rank-Nullity Theorem
• Null Space and Solution Sets of Ax=0
• Linear Independence
• Existence and Uniqueness of Solutions
• Basis and Dimension

Concept Map