Foundations — Cramer's rule

4.5.24 · D1 · Maths › Linear Algebra (Full) › Cramer's rule

Us sentence ka har letter aur symbol — ingredients, target, "box", "volume" — ab hum zero se build karenge. Is page par kuch bhi assumed nahi hai; parent note mein jo bhi notation use hui hai, woh yahan banai gayi hai, ek aisi order mein jahan har idea sirf pehle wale ideas par lean karta hai.

0. "System of equations" actually hai kya?

Koi bhi symbol se pehle, ek picture. Kaagaz par do seedhi lines khinchi hain. Har line ek equation hai — us line ka har point ek aisa pair $(x,y)$ hai jo us equation ko true banata hai. Jahan do lines cross karti hain woh ek aisa single pair hai jo dono ko true banata hai. Woh crossing point hi "solution" hai.

Figure s01 neeche exactly yahi draw karta hai: orange line ek equation hai, teal line doosri hai, aur plum dot us ek point $(2,1)$ ko mark karta hai jo dono par baitha hai — iska matlab hai "system solve karna."

Hamare liye "linear" kyun matter karta hai: seedhapan hi woh cheez hai jo poori determinant/volume machinery ko kaam karne deti hai. Curved equations har step ko tod deti.

1. Unknowns $x_1,x_2,\dots$ aur vector $\mathbf{x}$

Woh chhote numbers jinhe hum solve kar rahe hain woh unknowns hain. Hum inhe $x_1,x_2,\dots ,x_n$ likhte hain. Subscript sirf ek name-tag hai: $x_1$ matlab "pehla unknown", $x_2$ "doosra". 2D problem mein aap inhe plain $x$ aur $y$ ke roop mein dekh sakte hain.

Letter $n$ ka matlab sirf "jitne bhi unknowns hain utne" hai. Agar $n=2$ to do unknowns hain, agar $n=3$ to teen hain.

2. Matrix ke columns, aur $A$

Matrix ko column by column padho: har vertical strip khud ek vector hai — ek arrow. Yahi $A$ ko dekhne ka sabse zyada important tarika hai Cramer's rule ke liye.

Figure s02 ek $2\times 2$ grid leta hai aur uski do vertical strips ko origin se do arrows ke roop mein redraw karta hai: orange arrow column ${\mathbf{a}}_1$ hai, teal arrow column ${\mathbf{a}}_2$ hai. Picture kehti hai "matrix bas side by side khade arrows ka ek bundle hai."

Topic ko columns kyun chahiye: Cramer's rule poori tarah $A$ ke ek column ko swap karne ke baare mein hai. Agar aap $A$ ko arrows ki ek row ke roop mein nahi dekhte, to poora rule magic lagta hai.

3. Product $A\mathbf{x}$ — columns ka ek mix

Ab ek crucial reinterpretation. School aksar "row ko column se multiply karo" sikhata hai. Sahi hai — lekin hamare liye honest picture alag aur kaafi zyada useful hai:

Figure s03 yeh "tip-to-tail" recipe dikhata hai: orange arrow $x_1{\mathbf{a}}_1$ hai, phir uski tip se shuru hokar hum teal arrow $x_2{\mathbf{a}}_2$ draw karte hain; origin se final tip tak plum arrow poora product $A\mathbf{x}$ hai. Woh plum arrow woh hai jo mix produce karta hai.

Yahi wajah hai ki Cramer ek column swap karta hai aur kabhi row nahi: $A\mathbf{x}$ literally columns se bana hai.

4. Target vector $\mathbf{b}$, aur equation $A\mathbf{x}=\mathbf{b}$

Sab jodkar: $\underset{\text{ingredients}}{\underbrace{A}}\underset{\text{amounts}}{\underbrace{\mathbf{x}}}=\underset{\text{target}}{\underbrace{\mathbf{b}}}.$

Yeh compact line poora problem hai. Parent note mein har worked example iska ek instance hai.

5. Determinant kya hai? — signed area/volume

Yahan engine hai. Parent note baar baar kehta hai "$\det$ volume measure karta hai." Chaliye woh picture earn karte hain.

Figure s04 do column-arrows ${\mathbf{a}}_1$ (orange) aur ${\mathbf{a}}_2$ (teal) se spanned parallelogram ko shade karta hai; shaded plum area $|\det A|$ hai. Arrows ko wide karo ya tilt karo aur shaded area — determinant — unke saath change ho jaata hai.

Teen facts is volume ke baare mein hi Cramer ko chahiye — parent note inhe multilinear and alternating kehta hai:

Fact 3 se golden test milta hai: $\det A=0$ ka matlab hai columns squashed flat hain (woh full volume span nahi karte), isliye aap inhe ek general $\mathbf{b}$ tak uniquely mix nahi kar sakte. Exactly tab Cramer's rule forbidden hai.

In volumes ke hands-on formulas ke liye Determinants aur Cofactor Expansion dekho; deeper properties Multilinear and Alternating Maps mein hain.

6. Identity matrix $I$ aur basis vectors ${\mathbf{e}}_k$

Parent note ka proof $I$ se ek "clever matrix" build karta hai. Isliye humein $I$ chahiye.

Do properties jo proof use karta hai:

• $A{\mathbf{e}}_k={\mathbf{a}}_k$ — $A$ ko pure axis-arrow se multiply karna $A$ ka column $k$ pick out karta hai.
• $\det I=1$ — unit box ka volume $1$ hai (haari measuring stick).

7. Clever matrix $X_i$ — $I$ aur $\mathbf{x}$ se build kiya gaya

Parent note ka proof ek matrix introduce karta hai jise woh $X_i$ kehta hai. Yeh magic nahi hai — hum ise yahan build karte hain taaki kuch bhi define hone se pehle use na ho.

8. Swapped matrix $A_i$, aur identity $A{X}_i=A_i$

Ab $A$ ko hamare clever $X_i$ se multiply karo. Matrix multiplication column by column kaam karta hai: product $A{X}_i$ ka column $k$ hai $A$ times $X_i$ ka column $k$.

• Har wall column ke liye ($k\ne i$), $X_i$ ka woh column ${\mathbf{e}}_k$ hai, aur $A{\mathbf{e}}_k={\mathbf{a}}_k$ — $A$ ka original column seedha wapas aa jaata hai.
• Special column $i$ ke liye, $X_i$ ka woh column $\mathbf{x}$ hai, aur $A\mathbf{x}=\mathbf{b}$ (yahi haara system hai!).

Isliye product $A$ ke har original column ko rakhta hai siwaaye $i$-ve ke, jo $\mathbf{b}$ ban jaata hai — yeh precisely $A_i$ hai: $A{X}_i=A_i.$

9. Determinants lena, aur ratio $x_i=\frac{\det A_i}{\det A}$

Ab har ingredient table par hai. $A{X}_i=A_i$ ke dono sides ka determinant lo aur multiplicativity rule use karo.

Ise $A{X}_i=A_i$ par apply karo: $\det (A{X}_i)=\det A_i⟹\det A\cdot \det X_i=\det A_i.$

Yeh move kyun kaam karta hai: left side ek single product $A{X}_i$ hai, isliye multiplicativity uske determinant ko $\det A\cdot \det X_i$ mein split karta hai. Humne §7 mein already earn kiya tha $\det X_i=x_i$, isliye $\det X_i$ factor wahi unknown hai jo hum chahte hain. Substitute karo: $\det A\cdot x_i=\det A_i.$

Aakhir mein $\det A$ se divide karo (tabhi allowed hai jab $\det A\ne 0$): $x_i=\frac{\det A_i}{\det A}.$ Upar: swapped box ka volume. Neeche: original box ka volume. Divide karo → pour-amount. Division demand karta hai $\det A\ne 0$ — aap kabhi flat box se divide nahi kar sakte.

Prerequisite map

Practical alternative Gaussian Elimination, Matrix Inverse ke through reformulation, aur "$\det A\ne 0$ iff unique solution" ki equivalence Invertible Matrix Theorem mein bhi dekho.

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