Foundations — Gram-Schmidt orthogonalization — algorithm

4.5.35 · D1 · Maths › Linear Algebra (Full) › Gram-Schmidt orthogonalization — algorithm

Parent note Gram–Schmidt orthogonalization — algorithm padhne se pehle, tumhe har woh symbol aaona chahiye jo woh tumhare saamne phenkta hai. Neeche, har symbol ko teen cheezein milti hain: simple words → picture → topic ko yeh kyun chahiye. Yeh is tarah order kiye gaye hain ki har ek pichle par lean kare (aptly).

1. Ek vector, aur woh arrow jo woh banaata hai

Andar ke chhote numbers, $3$ aur $1$, components kehlaate hain. $v=(3,1)$ mein pehla component batata hai ki kitna right, doosra kitna upar.

Components ko specific numbers commit kiye bina naam dene ke liye, hum likhte hain $v=(v_1,v_2)$: yahan $v_1$ pehla component hai (kitna right) aur $v_2$ doosra component hai (kitna upar). Subscripts ke do uses ke baare mein warning — dhyan se padho:

• Jab vector ke neeche ka letter wohi hai jo poora vector hai (jaise $v_1,v_2$ andar $v=(v_1,v_2)$), subscript us ek vector ka component pick karta hai.
• Jab hum baad mein kai alag vectors ko $v_1,v_2,v_3,\dots$ likhte hain, subscript ek name tag hai jiska matlab hai "vector number 1, vector number 2, ..."

Context hamesha batata hai kaun sa: kya hum ek vector ke coordinates khol rahe hain, ya vectors ki ek family naam de rahe hain? Dono ko crystal-clear rakhne ke liye, yeh page $v_1,v_2$ components ke liye use karta hai aur poore-vector names ko neeche family discussion ke liye reserve karta hai.

2. Length, likha jaata hai $\Vert v\Vert$

For $v=(3,1)$ we have $v_1=3,v_2=1$, so $\Vert v\Vert =\sqrt{3^2+1^2}=\sqrt{10}$. Double bars $\Vert \Vert$ bas "andar wale ki length" hain — inhe kabhi absolute value ke single bars se confuse mat karo.

3. Inner product $⟨u,v⟩$ — sab kuch ka dil

Yeh poore chapter ka sabse important symbol hai, isliye hum ise dheere-dheere build karte hain.

Yeh specific formula perpendicularity kyun detect karta hai? Dono arrows ke beech ke opening ko naam dena helpful hai. Maan lo $\theta$ (Greek letter "theta") dono arrows ke beech ke angle ko represent karta hai, jahan unki tails milti hain — $\theta =0^{\circ }$ jab woh same direction point karte hain aur $\theta =90^{\circ }$ jab woh perfect corner banate hain. Woh naam haath mein aane ke baad, inner product secretly barabar hota hai $⟨u,v⟩=\Vert u\Vert \Vert v\Vert \cos \theta ,$ jahan $\cos \theta$ us angle ka cosine hai. Jab arrows perpendicular hote hain, $\theta =90^{\circ }$ aur $\cos 90^{\circ }=0$, jo poore product ko zero tak kheench laata hai. Tumhe Gram–Schmidt use karne ke liye yeh cosine formula nahi chahiye, lekin yeh explain karta hai kyun $⟨u,v⟩=0$ "right angle" ke barabar statement hai.

Fuller treatment ke liye Inner product spaces dekho; abhi ke liye, "matching components multiply karo aur add karo, zero matlab perpendicular" — bas itna hi chahiye.

4. Orthogonal aur orthonormal

Word split hota hai ortho (right-angle) + normal (unit length). Ek unit-length vector $e_k$ likha jaata hai; letter $e$ ek clean, normalized direction ka traditional naam hai. Orthonormal basis dekho.

5. Projection ${\mathrm{proj}}_u(v)$ — shadow

Yeh woh ek operation hai jo parent note "building block" kehta hai.

Formula (parent note mein poora derive kiya gaya) hai ${\mathrm{proj}}_u(v)=\frac{⟨v,u⟩}{⟨u,u⟩}u.$ Ise padho: "$v$ $u$ se kitna agree karta hai" divided by "$u$ khud se kitna agree karta hai", times direction $u$. Fraction bas ek number hai (ek scalar), aur $u$ ko us se multiply karna $u$ ko stretch ya shrink karta hai shadow ke foot tak pahunchne ke liye. Deep dive Orthogonal projection par available hai.

6. Linear independence

7. Recursion symbols: $\sum$, base case, aur $u_k=v_k-\dots$

Parent ka key formula ke do parts hain. Pehle, ek base case jo batata hai kahan se shuru karo: $u_1=v_1(\text{bilkul pehla vector pehle se "clean" hai — remove karne ke liye pehle kuch nahi}).$ Phir, har baad ke vector $k=2,3,\dots$ ke liye general step: $u_k=v_k-{\sum }_{j=1}^{k-1}\frac{⟨v_k,u_j⟩}{⟨u_j,u_j⟩}{u}_j.$ Base case ke bina tum kabhi shuru nahi kar sakte, kyunki sum ko pehle ke $u_j$'s ka exist karna zaroori hai. Notation ke do pieces ko unpack karna deserves karte hain.

Recursion word ka matlab hai "har naaya answer pehle answers use karke define hota hai": $u_1$ $u_2$ ko feed karta hai, aur $u_1,u_2$ $u_3$ ko feed karte hain, aur aage bhi. Tum $u_5$ compute nahi kar sakte pehle $u_1,\dots ,u_4$ liye bina.

8. Span aur "same subspace"

Yeh foundations topic ko kaise feed karte hain

Neeche ka map top to bottom padha jaata hai: simple arrow (vector) un do measurements ko support karta hai jo hum us par build karte hain — inner product (agreement) aur norm (length). Woh do orthogonality test aur unit length ki idea dete hain, jo orthonormal mein combine hote hain. Inner product aur norm saath projection (shadow) bhi build karte hain, jiska poora purpose yeh hai ki hum shadow subtract kar sakein. Aakhir mein, subtraction, $\sum$ loop, linear independence ki requirement, aur unchanged span ka promise sab Gram–Schmidt mein feed hote hain. Agar tumhara browser diagram render nahi kar sakta, toh woh sentence words mein wahi diagram hai.

Equipment checklist

Parent note kholne se pehle, pakka karo ki tum yeh sab out loud answer kar sako.

Connections

• Inner product spaces — jahan $⟨\cdot ,\cdot ⟩$ rehta hai aur uske full rules
• Orthogonal projection — shadow operation, depth mein
• Linear independence — input set ke liye entry ticket
• Orthonormal basis — woh finished object jo Gram–Schmidt return karta hai
• QR decomposition — jahan yeh foundations kaam aate hain
• Least squares — orthogonal directions par built ek application