Foundations — Orthogonal sets and orthonormal basis

4.5.34 · D1 · Maths › Linear Algebra (Full) › Orthogonal sets and orthonormal basis

Parent note ka mazaa lene se pehle, tumhe har ek mark jo woh likhta hai apna banana hai. Hum ek picture se shuruaat karke har ek cheez build karenge, ek aise order mein jahan koi bhi cheez tab tak nahi aati jab tak woh earn nahi ho jaati.

1. Arrow: ek vector $\mathbf{v}$ kya hota hai

Simple words mein. $v_1$ hai "kitna right," $v_2$ hai "kitna upar." Inhe stack karo aur tumne origin se ek arrow pin kar diya.

Picture. Corner point (origin) se shuru karo aur $v_1$ steps east chalo, phir $v_2$ steps north. Start se finish tak ka arrow hi vector hai.

Topic ko iska kyun zaroorat hai. Poora chapter vectors ke sets $\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_p\}$ ke baare mein baat karta hai. Agar "arrow = stack of numbers" automatic nahi hai, toh baad ke koi bhi formulas ka koi matlab nahi banega.

2. ${\mathbb{R}}^n$: wo space jisme arrows rehte hain

Picture. ${\mathbb{R}}^2$ kagaz ki ek flat sheet hai. ${\mathbb{R}}^3$ woh room hai jisme tum baithe ho. Bade $n$ ke liye ${\mathbb{R}}^n$ ko hum draw nahi kar sakte, lekin 2D mein pictures ke saath jo bhi rules hum prove karte hain woh sab hold karte hain — algebra karne ka yahi fayda hai.

Topic ko iska kyun zaroorat hai. Parent likhta hai "a set in ${\mathbb{R}}^n$." Iska matlab sirf itna hai "flat $n$-dimensional space mein arrows ka ek guccha."

3. Dot product $\mathbf{u}\cdot \mathbf{v}$ — show ka star

YEH tool kyun, aur koi nahi? Hume ek aisa number chahiye jo yeh bataye ki do arrows kitna same direction mein point karte hain. Ordinary multiplication sirf numbers par kaam karti hai; dot product exactly woh machine hai jo do arrows ko ek "agreement score" mein combine karne ke liye bani hai. Yeh sawaal ka jawaab deta hai ki "kya yeh do arrows aligned hain, opposed hain, ya perpendicular hain?"

Sign padhna — sabhi cases, koi gap nahi:

• $\mathbf{u}\cdot \mathbf{v}>0$ → angle $90^{\circ }$ se kam, arrows broadly agree karte hain.
• $\mathbf{u}\cdot \mathbf{v}=0$ → angle exactly $90^{\circ }$, arrows perpendicular hain. ⭐ Yahi woh case hai jis par poora chapter bana hai.
• $\mathbf{u}\cdot \mathbf{v}<0$ → angle $90^{\circ }$ se zyada, arrows broadly oppose karte hain.
• Agar koi bhi arrow zero vector ho, toh dot product formula se $0$ hoga, lekin koi angle nahi hota — ek degenerate case jise hum dhyaan mein rakhte hain.

Topic ko iska kyun zaroorat hai. "Orthogonal" ko define kiya gaya hai ${\mathbf{u}}_i\cdot {\mathbf{u}}_j=0$ se. Aur magic coordinate formula bas dot products ka ek dhera hai. Puri tour ke liye dekho Dot product and norms.

4. Length $\Vert \mathbf{v}\Vert$ — norm

Picture. Plane mein, $\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$ ek right triangle ka hypotenuse hai jiske legs $v_1$ aur $v_2$ hain. Pythagoras deta hai $\Vert \mathbf{v}\Vert =\sqrt{v_1^2+v_2^2}$.

Topic ko iska kyun zaroorat hai. "Unit vector" aur "normalize" length ke through define hote hain. Aur coordinate denominator ${\mathbf{u}}_j\cdot {\mathbf{u}}_j$ bas $\Vert {\mathbf{u}}_j{\Vert }^2$ hai.

5. Unit vector aur "normalize"

Length se divide kyun karte hain? Kyunki $\Vert \frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }\Vert =\frac{\Vert \mathbf{v}\Vert }{\Vert \mathbf{v}\Vert }=1$. Ek forbidden input hai $\mathbf{v}=0$: tum zero length se divide nahi kar sakte, aur zero arrow ki koi direction nahi hoti.

Topic ko iska kyun zaroorat hai. "Orthonormal" = orthogonal plus har vector normalized. Parent mein Example B literally ${\mathbf{q}}_1=\frac{1}{\sqrt{11}}{\mathbf{u}}_1$ build karta hai.

6. Perpendicular = orthogonal

Naya word kyun? "Perpendicular" 2D drawings se tied lagta hai. "Orthogonal" hume yaad dilata hai ki test ($\mathbf{u}\cdot \mathbf{v}=0$) kisi bhi dimension mein kaam karta hai jahan hum draw nahi kar sakte. Same concept, wider reach.

7. Linear combination, span, aur basis

Picture. Har $c_i{\mathbf{u}}_i$ arrow ${\mathbf{u}}_i$ ko $c_i$ se stretch karta hai; inhe tip-to-tail rakhne se tum ek naye point par pahunchte ho.

Topic ko iska kyun zaroorat hai. Chapter ki poori headline — "ek orthonormal basis ek dream coordinate system hai" — assume karti hai ki tum jaante ho ki ek basis sirf ek coordinate system hai, aur parent ka Section 2 prove karta hai ki orthogonal sets automatically independent hote hain.

8. Coordinates $c_j$

Picture. Coordinates "instructions" hain — $\mathbf{y}$ tak pahunchne ke liye har axis ke along kitna chalna hai. Ek generic basis ke liye tum inhe ek system solve karke paate ho; ek orthogonal ke liye tum inhe dotting se free mein paate ho. Yahi contrast is topic ka poora point hai. Ek aisi coordinate ke geometric meaning ke liye dekho Orthogonal projection.

9. Matrix, transpose, aur $U^{\mathsf{T}}U$

Ek orthonormal set ke liye woh grid diagonal par $1$ aur baaki jagah $0$ hota hai — identity matrix $I$. Yah ek fact QR decomposition aur Least squares ko power deta hai.

10. Yeh sab topic ko kaise feed karta hai

Baayein taraf sab kuch prerequisites hain; arrows dikhate hain ki har idea kya unlock karta hai. Parent topic neeche baithe hai, orthonormal bases aur identity $U^{\mathsf{T}}U=I$ se fed hoke.

Equipment checklist

Daayein side dhako aur khud ko test karo.