Foundations — Pivot positions, free variables

4.5.11 · D1 · Maths › Linear Algebra (Full) › Pivot positions, free variables

Hum "pivots" aur "free variables" ki baat karne se pehle, parent note mein jo bhi notation use ki gayi hai uska har piece earn karna hoga. Hum unhe ek-ek karke build karenge, har ek pichle ke upar. Kuch bhi assume nahi kiya gaya.

1. Unknowns $x_1,x_2,\dots ,x_n$ — aur subscript ka matlab

$x,y,z$ kyun nahi? Jab teen se zyada unknowns ho jaayein toh letters khatam ho jaate hain. $x_1,\dots ,x_n$ likhne se hum kitne bhi unknowns ke baare mein ek saath baat kar sakte hain, aur letter $n$ "unknowns ki total sankhya" ke liye khada hai.

2. Linear equation kya hota hai? (atom)

Ab jab ki $x_1,x_2,\dots$ ka matlab "labelled unknowns" hai, hum poore subject ki sabse chhoti cheez build kar sakte hain.

Picture — figure padhein. Neeche wala figure equation $2x_1+3x_2=7$ ko plot karta hai. Horizontal axis $x_1$ hai, vertical axis $x_2$ hai. Blue line un tamam pairs $(x_1,x_2)$ ka set hai jo equation ko sach banate hain; orange dots do example solutions hain jo uss par baithe hain. Dhyan dijiye ki line bilkul seedhi hai — yahi seedhapan hai jo "linear" hume deta hai.

Yeh topic isko kyun chahta hai. Poora subject inke systems ke baare mein hai — ek saath kai seedhi constraints. Agar equations modni ya curved hoti, toh koi bhi tidy counting rules kaam nahi karti. Seedhapan hi hai jo aage sab kuch (pivots, ranks, dimensions) kaam karta hai.

3. Letter $n$ — unknowns (columns) ko count karna

Picture. Apne unknowns ko left se right line up karein. $n$ slots hain. Baad mein, har slot numbers ka ek column ban jaata hai — isliye "$n$ columns" aur "$n$ variables" ek hi count niklengi. Woh fact yaad rakhein: yeh free-variable formula ka dil ban jaata hai jab woh pichla symbol define ho jaata hai jis par usay zaroorat hai.

4. Equations ko system mein stack karna aur grid ki tarah padhna

Woh idea jo sab kuch unlock karta hai: unknowns sirf positions mark karte hain. Hum $x$'s hat sakte hain aur sirf numbers rakh sakte hain, ek grid mein lined up:

$\begin{array}{cccc}1& 2& 1& 1\\ 2& 4& 3& 3\end{array}$

Har row ek equation hai. Left pe har column un numbers ko hold karta hai jo kisi ek particular unknown se multiply ho rahe hain. Numbers ka woh grid ek matrix hai.

5. Matrix kya hota hai? Rows, columns, aur size $m\times n$

Figure padhein. Figure hamara $2\times 3$ numbers ka grid dikhata hai. Left pe orange arrow do rows ko span karta hai aur labelled hai "$m=2$ rows (equations)". Upar green arrow teen columns ko span karta hai aur labelled hai "$n=3$ columns (variables)". In dono directions ko apne mind mein fix karein: rows across chalti hain, columns neeche jaati hain.

Yeh topic isko kyun chahta hai. Parent note ka har rule aise phrase kiya gaya hai ki "agar $A$ $m\times n$ hai aur $r$ pivots hain…". Free-variable formula tab tak use nahi kar sakte jab tak aap na bata sakein ki kaun sa count $m$ hai aur kaun sa $n$ hai. $n$ woh hai jo free variables ke liye matter karta hai kyunki free variables columns se aate hain.

6. Compact notation $A\mathbf{x}=\mathbf{b}$

$\mathbf{x}$ aur $\mathbf{b}$ mein bold signal karta hai ki "yeh numbers ka poora column hai", koi akela number nahi. Saada $A\mathbf{x}=\mathbf{b}$ ek saath poore system ka shorthand hai.

7. Augmented matrix $[A\mid \mathbf{b}]$

Yeh topic isko kyun chahta hai. Parent note ka consistency test — "kya augmented column ek pivot column hai?" — is glued-on last column ke baare mein ek statement hai. Aap woh sawaal tab hi pooch sakte hain jab $\mathbf{b}$ attached rakha ho.

8. Row reduction aur echelon form (borrowed tool — hum iska use kyun karte hain)

Teen legal moves (Row Reduction and Echelon Forms dekhein):

1. Do rows swap karo.
2. Ek row ko kisi nonzero number se multiply karo.
3. Ek row ka multiple doosri row mein add karo.

Figure padhein. Figure ek reduced grid dikhata hai. Red bold numbers leading entries hain — har nonzero row ka pehla nonzero number — aur blue staircase line trace karti hai ki har ek pichle se down-and-right kaise step karta hai. Column 2 (green note) kabhi kisi leading number ka host nahi banta; uss par nazar rakho, yeh hamara pehla free column ban jaata hai.

9. Leading entry, pivot position, pivot column — payoff terms

Ab parent ki core definitions mein har symbol earn ho gaya hai.

Picture. Figure 3 ke staircase mein, pivot woh "corner" number hai jahan har step shuru hota hai. Uska column ek particular unknown solve karne ke liye committed hai; ek non-pivot column woh column hai jis par koi step kabhi nahi uta.

10. Rank $r$ — sirf "kitne pivots hain"

Yeh topic isko kyun chahta hai. Kyunki "pinned-down unknowns ki sankhya" $=$ "pivots ki sankhya" $=r$, aur "leftover free unknowns" $=n-r$. Poora counting formula $r$ ko pivot count maanne par hi jeeta-marta hai.

11. Consistency — kya koi solution hai bhi?

Count karne se pehle kyun aata hai. Free variables solution set ka size measure karte hain — lekin sirf tabhi jab solution exist karta ho. Yeh woh alag check hai jiske baare mein parent note bar bar warning deta hai, Existence and Uniqueness of Solutions mein expand kiya gaya hai. (Section 6 se yaad karein ki ek homogeneous system $A\mathbf{x}=0$ kabhi is trap mein nahi hai — woh hamesha consistent hai.)

12. Count ko kaam mein lagana — $n-r$ ka matlab kya hai

Yahan woh payoff hai jo har symbol ko ek saath bandh karta hai. Maano system consistent hai (warna zero solutions hain, bas khatam). Toh:

Toh logic chain yeh hai: consistent? → agar nahi, zero solutions; agar haan, $n-r$ dekho → $0$ matlab unique, $>0$ matlab infinitely many. Woh ek akela number poora qualitative outcome decide karta hai.

13. Bracket-vector solution form (woh $\left[\begin{array}{c}\cdots \end{array}\right]+t\left[\begin{array}{c}\cdots \end{array}\right]$ notation)

Parent ka Example 1 ka answer tha $\mathbf{x}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]+t\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right].$

Figure padhein. 3-D figure is solution ko plot karta hai. Orange dot fixed start point $(0,0,1)$ hai. Green arrow direction $(-2,1,0)$ hai jis par aap slide karte hain jab knob $t$ ghoomate hain. Blue line solutions ka poora set hai — uss par har point kisi $t$ ki value ke liye ek valid $\mathbf{x}$ hai. (Null Space and Solution Sets of Ax=0 mein aur gehraai se.)

Prerequisite map

Neeche wala diagram ek flow chart hai: har box is page ka ek concept hai, aur har arrow ka matlab hai "tail wala box head wale box se pehle samajh mein aana chahiye." Ise top to bottom padhein — "Linear equation" se shuru karein aur arrows follow karein; har path eventually neeche wale parent topic mein funnel ho jaata hai. Yeh upar build kiye hue dependency order ka sirf ek visual table of contents hai.

Equipment checklist

Har line reveal karein aur check karein ki aap bina hesitation ke answer de sakte hain.

Connections

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