4.5.19
Ek vector v pehle exist karta hai. Coordinate vector woh numbers ki list hai jo tum usi arrow ko describe karne ke liye likhte ho jab ek ruler-system (ek basis) choose kar liya ho. Basis badlo → arrow apni jagah rehta hai, lekin numbers badal jaate hain. Change of basis woh dictionary hai jo ek hi arrow ke liye ek list of numbers ko doosri list mein translate karti hai.
Definition Coordinate vectorMaano B = { b 1 , … , b n } V v ∈ V unique expansion hota hai:
v = c 1 b 1 + c 2 b 2 + ⋯ + c n b n . v coordinate vector relative to B [ v ] B = c 1 ⋮ c n .
Yeh unique KYU hai? Kyunki B linearly independent hai. Agar do expansions hote, unhe subtract karne par ek non-trivial combination 0 asli wajah hai jis se coordinates well-defined hote hain.
Tumhe [ v ] B C [ v ] C P [ v ] C = C ← B P [ v ] B .
[ v ] B matlab kya hai, wahan se shuru karo:
v = c 1 b 1 + ⋯ + c n b n .
Ab dono sides ke coordinates C [ ⋅ ] C linear hai (neeche prove kiya gaya hai), isliye:
[ v ] C = c 1 [ b 1 ] C + ⋯ + c n [ b n ] C .
Right side bilkul ek matrix times c 1 ⋮ c n [ b j ] C
Coordinate maps linear kyun hote hain: u = ∑ a i b i w = ∑ d i b i u + α w = ∑ ( a i + α d i ) b i [ u + α w ] B = [ u ] B + α [ w ] B
Roles swap karne par B ← C P B ← C P = ( C ← B P ) − 1 . Kyun? C B I
R n E B = [ b 1 ⋯ b n ] E ← B P = B , B ← E P = B − 1 . B B B − 1 C ← B P = C − 1 B .
Worked example Example 1 — basicR 2 B = { b 1 , b 2 } b 1 = [ 1 1 ] , b 2 = [ 1 − 1 ] v = [ 4 2 ] [ v ] B
Humein c 1 b 1 + c 2 b 2 = v B [ v ] B = v [ v ] B = B − 1 v Yeh step kyun? Standard basis mein v B − 1 B B = [ 1 1 1 − 1 ] , B − 1 = − 2 1 [ − 1 − 1 − 1 1 ] = 2 1 [ 1 1 1 − 1 ] . [ v ] B = 2 1 [ 1 1 1 − 1 ] [ 4 2 ] = [ 3 1 ] . 3 b 1 + 1 b 2 = [ 3 3 ] + [ 1 − 1 ] = [ 4 2 ] .
Worked example Example 2 — do non-standard bases ke beechB C = { c 1 , c 2 } c 1 = [ 1 0 ] , c 2 = [ 1 2 ] C ← B P
P = C − 1 B Kyun? P [ b j ] C = C − 1 b j C − 1 B C = [ 1 0 1 2 ] , C − 1 = [ 1 0 − 2 1 2 1 ] . P = C − 1 B = [ 1 0 − 2 1 2 1 ] [ 1 1 1 − 1 ] = [ 2 1 2 1 2 3 − 2 1 ] . = [ b 1 ] C 2 1 c 1 + 2 1 c 2 = 2 1 [ 1 0 ] + 2 1 [ 1 2 ] = [ 1 1 ] = b 1
Worked example Example 3 — Forecast-then-Verify (polynomials)V = P 2 B = { 1 , t , t 2 } C = { 1 , 1 + t , 1 + t + t 2 } p ( t ) = 2 + 3 t + t 2 Forecast: C a ( 1 ) + b ( 1 + t ) + c ( 1 + t + t 2 ) = 2 + 3 t + t 2
t 2 c = 1 Kyun? Sirf c 3 t 2 t b + c = 3 ⇒ b = 2 const: a + b + c = 2 ⇒ a = − 1 [ p ] C = − 1 2 1 negative hai — coordinates positive hone zaroori nahi.
P naye basis vectors c j Kyun sahi lagta hai: tum C mein ja rahe ho, toh lagta hai C Fix: columns purane vectors b j naye coordinates mein expressed: [ b j ] C B B
C − 1 B B − 1 C Kyun sahi lagta hai: dono letters aate hain; order arbitrary lagta hai. Fix: units check karo. C ← B P = C − 1 B B B C − 1 C
Common mistake Yeh sochna ki vector khud badal jaata hai.Kyun sahi lagta hai: numbers clearly badal jaate hain. Fix: sirf description badlti hai. v [ v ] C = P [ v ] B
Recall Quick self-test (answers cover karo)
Q: C ← B P [ b j ] C
Q: R n B , C C ← B P C − 1 B
Q: C ← B P B ← C P
Q: [ v ] B
Coordinate vector [ v ] B Scalars c i v = ∑ c i b i
Coordinates unique kyun hote hain? Kyunki basis linearly independent hoti hai, isliye expansion unique hoti hai.
Change-of-basis matrix P C ← B Purane basis vectors naye basis mein expressed: [ b j ] C
R n B , C P C ← B = ? C − 1 B
P C ← B P B ← C Yeh ek doosre ke inverses hain.
Kya basis change karne se vector badalta hai? Nahi — sirf uski coordinate description badlti hai; vector invariant rehta hai.
Coordinate map linear kyun hai? Vectors ko add/scale karne par unke B
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek treasure map par ek toy rakha hai. Ek dost door se maap karta hai, doosra khidki se. Toy nahi hila, lekin dono doston ne alag-alag "steps left, steps up" numbers likhe. Change of basis woh chota rulebook hai jo ek dost ke numbers doosre ke numbers mein turn karta hai — sirf directions translate karta hai, toy kabhi nahi hilata.
"New columns of Old, then C-inverse-B."
Columns = naye kapdon mein old vectors b j C − 1 B C − 1 C B B
Basis and dimension — coordinates tabhi exist karte hain jab basis fix ho.Linear independence — coordinate uniqueness guarantee karta hai.Invertible matrices — B , C Similar matrices and diagonalization — A ′ = P − 1 A P operators ke liye change of basis hai.Linear transformations and their matrices — coordinate maps abstract maps ko concrete banate hain.
old vectors in new coords
Columns are b_j in C coords
Standard basis shortcut P equals B
