Change of basis matrix

4.5.20 · Maths › Linear Algebra (Full)

1. Setup — Hum actually kya translate kar rahe hain?

Change of basis matrix KYA hai? Do bases $B$ (purani) aur $C$ (nayi) diye hue, yeh matrix $P_{C \leftarrow B}$ hai jaise ki $[v]_{C} = P_{C \leftarrow B} [v]_{B} .$ Yeh $B$ -coordinates andar leta hai aur $C$ -coordinates bahar deta hai.

2. First principles se Derivation

Hum ek aisi matrix banana chahte hain jo, $[v]_{B}$ ko feed karne par, $[v]_{C}$ return kare. Chaliye ise build karte hain.

Step 1 — $v$ ko purani basis mein likho. Definition se, $v = c_{1} b_{1} + \dots + c_{n} b_{n} .$ Yeh step kyun? Yahi ek cheez hum jaante hain: $[v]_{B} = (c_{1}, \dots, c_{n})$ .

Step 2 — Har purane basis vector ko nayi basis mein express karo. Har $b_{j}$ khud ek vector hai, isliye uske $C$ -coordinates hain: $[b_{j}]_{C} = p_{1 j} ⋮ p_{nj}, i.e. b_{j} = \sum_{i} p_{ij} c_{i} .$ Yeh step kyun? Translation "term by term" hoti hai: agar mujhe pata hai ki har building block $b_{j}$ ko nayi language mein kaise kehte hain, toh main koi bhi combination keh sakta hoon.

Step 3 — Pure sum ke $C$ -coordinates lo. Coordinates linear hain, isliye $[v]_{C} = [\sum_{j} c_{j} b_{j}]_{C} = \sum_{j} c_{j} [b_{j}]_{C} .$ Yeh step kyun? Coordinate map $v \mapsto [v]_{C}$ ek linear map hai (expansion ki uniqueness yeh guarantee karti hai), isliye yeh sums aur scalars par distribute hota hai.

Step 4 — Matrix–vector product ko pehchano. $[b_{j}]_{C}$ ko columns ke roop mein stack karo: $[v]_{C} = P_{C \leftarrow B} [[b_{1}]_{C} [b_{2}]_{C} \dots [b_{n}]_{C}] c_{1} ⋮ c_{n} = P_{C \leftarrow B} [v]_{B} . ■$

3. Standard-Basis Shortcut

$R^{n}$ mein standard basis $E = {e_{1}, \dots, e_{n}}$ ke saath:

$P_{E \leftarrow B} = [b_{1} ∣ b_{2} ∣ \dots ∣ b_{n}]$ — bas basis vectors ko columns ke roop mein rakh do! (Kyunki $[b_{j}]_{E} = b_{j}$ literally hota hai.) Ise $B$ bolte hain (notation ka abuse: basis vectors ki matrix).
Doosri taraf: $P_{B \leftarrow E} = B^{- 1}$ , kyunki translate karke wapas translate karna identity hai: $P_{B \leftarrow E} P_{E \leftarrow B} = I .$

4. Worked Examples

5. Common Mistakes (Steel-manned)

6. Active Recall

Recall

$P_{C \leftarrow B}$ ka $j$ -waan column kya hai? Purana basis vector $b_{j}$ jo $C$ -coordinates mein likha gaya hai: $[b_{j}]_{C}$ .

Recall Agar

$B, C$ basis vectors ki matrices hain (standard coords), toh $P_{C \leftarrow B}$ kya hai? $P_{C \leftarrow B} = C^{- 1} B$ .

Recall Change of basis ko reverse kaise karte hain?

Inverse lo: $P_{B \leftarrow C} = (P_{C \leftarrow B})^{- 1}$ .

Recall (Feynman, ek 12-saal ke bachche ko explain karo)

Socho ek khaazana ek park mein rakha hai. Ek dost directions deta hai "north/east steps" use karke, doosra "oak-ki-taraf / pond-ki-taraf steps" use karke. Khaazana kabhi move nahi karta! Change of basis matrix ek chhoti si translator card hai: tum use ek dost ki language mein steps batao, woh tumhe usi jagah ke steps doosre dost ki language mein batata hai. Kaunsa dost bol raha hai isko swap karne ke liye, card ko palat do (yahi inverse hai).

7. Connections

Basis and Dimension — coordinates ke exist karne ke liye basis chahiye.
Coordinate Vectors — woh columns jinhe hum manipulate karte hain.
Invertible Matrices — change of basis matrix hamesha invertible hoti hai (bases independent hain).
Similar Matrices — $A^{'} = P^{- 1} A P$ ek linear map ko nayi basis mein rewrite karne ke liye change of basis use karta hai.
Eigenvectors and Diagonalization — diagonalizing = eigenbasis mein change karna.
Linear Transformations — active (vectors move hote hain) vs passive (description badlti hai) views.

P_{C \leftarrow B}

ek coordinate vector ke saath kya karta hai?

B

-coordinates input leta hai aur

C

-coordinates return karta hai:

[v]_{C} = P_{C \leftarrow B} [v]_{B}

Change of basis matrix

P_{C \leftarrow B}

ka column

j

kya hai?

[b_{j}]_{C}

, purana basis vector

b_{j}

nayi basis

C

mein express kiya gaya.

Agar

B, C

mein columns basis vectors hain (standard coords mein), toh

P_{C \leftarrow B} =

C^{- 1} B

Standard coords ko basis

B

coords mein kaise translate karein?

B^{- 1}

se multiply karo:

[v]_{B} = B^{- 1} v

Basis

B

coords ko standard mein wapas kaise translate karein?

B

se multiply karo:

[v]_{E} = B [v]_{B}

(

B

ke columns standard coords mein basis vectors hain).

P_{C \leftarrow B}

ko reverse kaise karte hain?

Iska inverse use karo

P_{B \leftarrow C} = (P_{C \leftarrow B})^{- 1}

Kya change of basis vector ko move karta hai?

Nahi — vector fixed hai; sirf uski coordinate description badlti hai (passive view).

Har change of basis matrix invertible kyun hoti hai?

Iske columns ek basis ke coordinates hain (independent, spanning), isliye yeh full rank wali square matrix hai.