Orthogonal sets and orthonormal basis — Hinglish notes

4.5.34 · Maths › Linear Algebra (Full)

1. Definitions, carefully build ki gayi

DIFFERENCE kya hai? Orthogonal = sirf perpendicular. Orthonormal = perpendicular aur length 1. Aap ek orthogonal set ko orthonormal banate ho har vector ko uski apni length se divide karke (normalizing).

2. WHY orthogonal sets automatically independent hote hain (derive karo)

Derivation from scratch. Maano $c_{1} u_{1} + c_{2} u_{2} + \dots + c_{p} u_{p} = 0 .$ Dono sides ka dot product ek fixed vector $u_{j}$ ke saath lo: $u_{j} \cdot (\sum_{i} c_{i} u_{i}) = u_{j} \cdot 0 = 0.$ Dot product distribute karo: $\sum_{i} c_{i} (u_{j} \cdot u_{i}) = 0.$ Yeh step kyun? Orthogonality ki wajah se, $i = j$ ke alawa har term zero ho jaati hai (woh dot products $0$ hain). Jo bachta hai: $c_{j} (u_{j} \cdot u_{j}) = c_{j} ∥ u_{j} ∥^{2} = 0.$ Kyunki $u_{j} \neq = 0$ hai, $∥ u_{j} ∥^{2} > 0$ hai, jo $c_{j} = 0$ force karta hai. Yeh har $j$ ke liye hold karta hai, toh saare coefficients zero hain → independent. $■$

3. THE payoff: dot product se coordinates (derive karo)

HOW hum yeh paate hain (first principles). Kyunki yeh ek basis hai, kuch coefficients exist karte hain: $y = c_{1} u_{1} + \dots + c_{p} u_{p} .$ Abhi tak hume $c_{i}$ pata nahi. Dono sides ka dot $u_{j}$ ke saath lo: $y \cdot u_{j} = \sum_{i} c_{i} (u_{i} \cdot u_{j}) .$ Yeh step kyun? Wahi purani trick — orthogonality saare cross terms nuke kar deti hai, sirf $c_{j} (u_{j} \cdot u_{j})$ bachta hai: $y \cdot u_{j} = c_{j} (u_{j} \cdot u_{j}) \Rightarrow c_{j} = \frac{y \cdot u _{j}}{u _{j} \cdot u _{j}} .$ Har coordinate decoupled hai — koi system solve nahi karna. Yahi poora magic hai.

Figure — Orthogonal sets and orthonormal basis

4. Orthogonal matrices

WHY $U^{T} U = I$ ? $U^{T} U$ ki $(i, j)$ entry exactly $u_{i} \cdot u_{j}$ hai. Orthonormality kehti hai yeh diagonal par $1$ hai, bahar $0$ — yaani identity.

5. Worked examples

6. Common mistakes (Steel-manned)

7. Active recall

Orthogonal set kya hota hai?

Vectors ka ek set jahan har distinct pair ka dot product 0 ho (

u_{i} \cdot u_{j} = 0

for

i \neq = j

Set ko orthonormal banane ki extra condition kya hai?

Har vector ek unit vector ho (

∥ u_{i} ∥ = 1

), yaani

u_{i} \cdot u_{i} = 1

Nonzero orthogonal set automatically linearly independent kyun hota hai?

\sum c_{i} u_{i} = 0

u_{j}

ke saath dot karne par

c_{j} ∥ u_{j} ∥^{2} = 0

bachta hai, jo

c_{j} = 0

force karta hai.

Orthogonal basis mein coordinate formula?

c_{j} = \frac{y \cdot u _{j}}{u _{j} \cdot u _{j}}

Orthonormal basis mein coordinate formula?

c_{j} = y \cdot u_{j}

(denominator 1 hai).

Orthonormal columns wali matrix U ke liye

U^{T} U

kya hai?

Identity

I

U^{- 1} = U^{T}

kab hold karta hai?

Jab U orthonormal columns ke saath square ho (ek orthogonal matrix).

Orthonormal transforms length preserve kyun karte hain?

∥ U x ∥^{2} = x^{T} U^{T} U x = x^{T} x = ∥ x ∥^{2}

Orthogonal set ko orthonormal mein kaise badlate ho?

Normalize karo: har vector ko uske apne norm se divide karo.

"Orthogonal matrix" ke saath common misnomer trap?

Ismein actually orthonormal (unit) columns aur square shape chahiye.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho arrows alag-alag directions mein point kar rahe hain jo saare perfect right angles par hain, jaise ek room ke edges (left-right, forward-back, up-down). Agar koi kahin khada hai aur aap batana chahte ho woh kahan hai, toh aapko koi puzzle solve nahi karna — bas poochho "floor ke saath kitna aage?" aur "kitna upar?" alag-alag, kyunki directions ek doosre mein interfere nahi karte. Dot product exactly wahi "is arrow ke saath kitna door" measurement hai. Orthonormal matlab bhi yeh hai ki har arrow exactly ek step lamba hai, toh tumhare answers already sahi units mein hain. Isliye yeh duniya ka sabse aasaan coordinate system hai.

Connections

Dot product and norms — yahan har cheez ke peeche measuring tool.
Gram-Schmidt process — kisi bhi basis se orthogonal/orthonormal basis kaise banate hain.
Orthogonal projection — Section 3 ka formula literally projections ka sum hai.
Least squares — orthonormal bases normal equations ko trivial bana dete hain.
QR decomposition — $Q$ ke orthonormal columns hain; $R$ coordinates store karta hai.
Eigenvalues and eigenvectors — symmetric matrices ke orthonormal eigenbases hote hain (Spectral Theorem).