Visual walkthrough — Systems of linear equations — matrix form Ax = b

4.5.8 · D2 · Maths › Linear Algebra (Full) › Systems of linear equations — matrix form Ax = b

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Step 1 — "Linear equation" asal mein hoti kya hai?

KYA HAI. Do unknown numbers par sabse simple balance condition se shuru karo, unhe $x$ aur $y$ kaho:

$$\underset{\text{how much }x\text{ counts}}{\underbrace{a}}x+\underset{\text{how much }y\text{ counts}}{\underbrace{b}}y=\underset{\text{the target total}}{\underbrace{c}}.$$

Yahan $a$, $b$, $c$ known fixed numbers hain; $x$, $y$ woh unknowns hain jinhein hum dhundh rahe hain. "Linear" ka matlab hai ki unknowns seedhe aate hain — kabhi square nahi, kabhi aapas mein multiply nahi, kabhi kisi curvy cheez ke andar nahi.

YE YAHAN SE KYUN SHURU KAREIN. Kuch bhi matrix mein pack karne se pehle, humein pata hona chahiye ki EK equation picture mein kaisi dikhti hai. Baad mein sab kuch isi ka stack hai.

PICTURE. Saare pairs $(x,y)$ jo ek aisi equation satisfy karte hain woh ek seedhi line banate hain. Yahi wajah hai "linear" shabd ki.

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Step 2 — Do equations = do lines = "row view"

KYA HAI. Ab do balance conditions ek saath maango:

$$\begin{array}{rl}{a}_{11}x+a_{12}y& =b_1(\text{line 1})\\ a_{21}x+a_{22}y& =b_2(\text{line 2})\end{array}$$

Subscripts ko seat number ki tarah padho: $a_{21}$ matlab row 2, column 1 — "2nd equation mein 1st unknown ka coefficient". Letter $b_1$ equation 1 ka target hai.

KYUN. Ek akeli equation ke answers ki puri line hoti hai — bahut zyada. Doosri equation add karna hi cheezein pin down karne ka tarika hai: ek point dono lines par ek saath hona chahiye.

PICTURE. Plane mein do lines. Solution unka single crossing point hai. Yeh row view hai: system ki har row ek line hai (3D mein, ek flat plane; fancy word hai hyperplane), aur solve karna = unhe intersect karna.

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Step 3 — Numbers pack karo: $A$, $x$, $b$ ka janam

KYA HAI. Ab hum sirf bookkeeping karte hain — koi nayi maths nahi, bas saaf-suthra storage. Chaar coefficients ko ek grid mein, do unknowns ko ek stack mein, do targets ko ek stack mein rakho:

$$A=\underset{\text{coefficient grid}}{\underbrace{\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]}},x=\underset{\text{unknowns}}{\underbrace{\left[\begin{array}{c}x\\ y\end{array}\right]}},b=\underset{\text{targets}}{\underbrace{\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]}}.$$

Ek matrix bas named slots waali numbers ki rectangle hai; ek vector yahan numbers ka ek column hai. Hum product $Ax$ ko define karte hain taaki use stack karke $b$ ke barabar rakhne se hamare do equations exactly mil jayein:

$$Ax=\left[\begin{array}{c}{a}_{11}x+a_{12}y\\ a_{21}x+a_{22}y\end{array}\right]\stackrel{!}{=}\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=b.$$

Chhota $\stackrel{!}{=}$ matlab "hum chahte hain ki yeh hold kare". $Ax$ ki row $i$, $A$ ki row $i$ aur $x$ ka dot product hai — matching entries multiply karo, add karo. Dekho Matrix multiplication.

AISE DEFINE KYUN KAREIN. Kyunki hum chahte hain ki compressed symbol $Ax=b$ ka matlab wahi ho jo do lambi equations ka hai. Product ki definition ko reverse-engineer kiya gaya hai taaki yeh sach ho — isse zyada mysterious kuch nahi.

PICTURE. Grid, do stacks, aur ek arrow jo dikhata hai ki row 1 kaise $x$ ke "across" pahunchkar equation 1 banati hai.

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Step 4 — $Ax$ ko COLUMNS se re-slice karo: arrows mix karna

KYA HAI. Same product $Ax$, alag tarike se kata. Rows ke across padhne ki jagah, woh terms collect karo jo $x$ carry karte hain aur jo $y$ carry karte hain:

$$Ax=\left[\begin{array}{c}{a}_{11}x+a_{12}y\\ a_{21}x+a_{22}y\end{array}\right]=x\underset{\text{column }1}{\underbrace{\left[\begin{array}{c}{a}_{11}\\ a_{21}\end{array}\right]}}+y\underset{\text{column }2}{\underbrace{\left[\begin{array}{c}{a}_{12}\\ a_{22}\end{array}\right]}}.$$

Columns ko $a_{(1)}=\left[\begin{array}{c}{a}_{11}\\ a_{21}\end{array}\right]$ aur $a_{(2)}=\left[\begin{array}{c}{a}_{12}\\ a_{22}\end{array}\right]$ kaho. Phir

$$Ax=x{a}_{(1)}+y{a}_{(2)}.$$

Unknowns $x$ aur $y$ scoop counts ban gaye — arrow $a_{(1)}$ ke kitne scoops, arrow $a_{(2)}$ ke kitne scoops.

YEH RE-SLICE KYUN MAAYANE RAKHTI HAI. Yeh algebra ko geometry mein convert karti hai: $Ax$ column arrows ki ek linear combination hai. $Ax=b$ solve karna ek visual question ban jaata hai — "har column arrow ke kitne scoops add karun taaki main exactly target arrow $b$ par land karun?" Yeh column view hai, aur yeh poore chapter ka 80/20 hai.

PICTURE. Do column arrows origin se shuru hote hain; har ek ko stretch/shrink karo, phir tip-to-tail add karo. Woh scoop counts $(x,y)$ jo sum ko $b$ tak pahunchate hain woh solution hai.

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Step 5 — Existence rule: kya $b$ reachable hai?

KYA HAI. $Ax=b$ tab aur sirf tab solution rakhta hai jab target $b$ us reachable set mein kahin ho:

$$Ax=b\text{ solvable}⟺b\in \mathrm{Col}(A).$$

KYUN. Agar columns ke scoops ka koi bhi combination kabhi $b$ ki taraf point nahi karta, toh koi $x$ system solve nahi karta — bas. Columns us arrow ko simply manufacture nahi kar sakte.

PICTURE. Jab do columns alag directions mein point karte hain toh woh poora plane fill kar dete hain, isliye har $b$ reachable hai (left). Jab woh parallel hain, toh woh sirf ek line ke along pahunch sakte hain — us line se bahar ka $b$ unreachable hai (right). "Alag directions" Linear independence ka visual meaning hai.

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Step 6 — Transformation view: kaun sa input $b$ par land karta hai?

KYA HAI. $A$ ko ek machine $x\mapsto Ax$ ki tarah padho: ek arrow andar daalo, ek arrow bahar aata hai. $Ax=b$ solve karna machine ka sawaal ulta poochta hai: machine konsa input arrow $b$ tak bhejti hai?

KYUN. Yeh lens uniqueness explain karta hai. Agar machine plane ko ek line par squash kar de (parallel columns), toh bahut alag inputs same output par flatten ho jaate hain — toh ek reachable $b$ ke bahut saare preimages hote hain. Agar machine plane ko poora aur 2D rakhti hai, toh har output ka exactly ek input hota hai.

PICTURE. Left par input arrows ka ek grid; machine unhe right par output arrows mein warp karti hai. Us input ko highlight karo jo $b$ par map karta hai.

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Step 7 — Teen verdicts (saare cases cover hain)

KYA HAI. Sab kuch combine karo. $r=\mathrm{rank}(A)$ = woh genuinely independent directions ki count jo columns dete hain (dekho Rank of a matrix), $n$ = unknowns ki count, aur augmented grid $[A|b]$ se compare karo ($b$ ko ek extra column ki tarah glue karo):

Picture	Condition	Verdict
Lines ek point par cross karti hain	$\operatorname{rank}A=\operatorname{rank}[A,	,b]=n$
Ek hi line twice	$\operatorname{rank}A=\operatorname{rank}[A,	,b]<n$
Parallel, kabhi nahi milti	$\operatorname{rank}A<\operatorname{rank}[A,	,b]$

KYUN. $b$ glue karke rank jump dhundna matlab $b$ ne ek fresh direction maangi jo columns supply nahi kar sakte → inconsistent. Agar ranks match karti hain lekin unknowns se kam independent equations hain, toh leftover unknowns free float karte hain (unki count $n-r$ hai).

PICTURE. Teen canonical row pictures side by side.

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Ek picture mein summary

Neeche, SAME product $Ax$ ko teeno tarikon se padha gaya hai — rows cross karti hain, columns mix hoti hain, machine map karti hai — sab ek hi answer par agree karte hain. Yeh agreement hi linear algebra hai.

Recall Feynman retelling — kisi dost ko batao

Tumhare paas do dials hain, $x$ aur $y$. Har dial control karta hai ki ek ingredient arrow ($A$ ka ek column) ke kitne scoops tum ek bowl mein daalte ho. Daale gaye arrows ko tip-to-tail add karne par ek final arrow milta hai. Recipe yeh poochti hai: dials aise set karo ki final arrow exactly target $b$ par land kare. Agar do ingredient arrows alag directions mein point karte hain, toh woh plane mein kuch bhi bana sakte hain, isliye hamesha exactly ek dial setting hoti hai — unique. Agar dono ingredients ek hi direction mein point karte hain, toh tum sirf us ek line ke along arrows hi bana sakte ho: line se bahar ka target impossible hai (no solution), jabki line par ka target endless taaron se banaya ja sakta hai (ek dial upar karo aur doosre ko neeche karke compensate karo). Inhi numbers ko crossing lines (row view) ya squashing machine (transformation view) ki tarah padhna identical story batata hai — isliye $Ax=b$ ek line mein poora subject hai.

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Connections

• Matrix multiplication — row-dot-column rule jo $Ax$ define karta hai (Step 3).
• Column space and null space — reachable set aur floating directions (Steps 4–5).
• Rank of a matrix — independent directions count karta hai; verdict decide karta hai (Step 7).
• Linear independence — "columns alag directions mein point karti hain" precisely (Step 5).
• Gaussian elimination — woh machine jo ranks aur scoops compute karti hai.
• Determinant — square case mein test ki "columns plane fill karte hain".
• Inverse of a matrix — unique case ka formula $x=A^{-1}b$.