4.5.15 · D1Linear independence — formal definition, testing
Arrows ka ek collection linearly independent tab hota hai jab har arrow genuinely naya direction point kare jo baaki koi arrow stretch aur add karke reach nahi kar sakta. Is topic mein jo bhi hai — vectors, scalars, equation A c = 0 "kya koi arrow baaki arrows ki leftover copy hai?"
Yeh page har woh notation build karta hai jo parent note use karta hai, bilkul zero se. Ise upar se neeche padho: har item agli item mein use hota hai.
Ek vector ek aisa arrow hai jisme length aur direction hoti hai, ek starting point (origin) se draw kiya hua. Hum ise numbers ki list ke roop mein bhi likhte hain — iske coordinates — jo batata hai ki har axis ke saath kitna aage jaana hai.
Dono views ek hi object hain. List ( 3 , 2 ) 3 x 2 y
Numbers ki list kyun?
Arrows draw karna ek picture ke liye theek hai, lekin compute karne ke liye numbers chahiye. List woh arrow hai jo calculator ki samajh mein aane wali language mein likha gaya hai. Is topic mein hum in lists ko side by side rakhkar ek table (matrix) banate hain, isliye number-list view zaroori hai.
Hum vectors ko bold mein likhte hain: v 1 , v 2 , … name tag hai (vector number 1, number 2), koi power ya coordinate nahi.
v 2 v Yeh sahi kyun lagta hai: subscripts exponents jaisi dikhti hain.
Fix: v 2 label hai — hamare list mein doosra vector. Iska v
Ek scalar ek ordinary single number hota hai (jaise 2 − 2 1 0 scale se aata hai: kisi vector ko scalar se multiply karna use stretch ya shrink karta hai, aur negative scalar use ulti direction mein flip karta hai.
Parent note scalars ke liye letters c 1 , c 2 , … , c n c i v i amount hai jo hum lete hain.
Intuition Is topic ke liye scalars kyun chahiye"Kya vector A baaki se build ho sakta hai?" ka matlab hai "kya main baaki ko scale karke add kar sakta hoon taaki A pe land karun?" Scaling recipe ka aadha hissa hai. Scalars c i
Special values dhyan se dekho:
c = 1 c = 0 kuch bhi nahi lete.c = − 1 c = 2
Definition Vector addition (tip-to-tail)Do vectors add karne ke liye, doosre ka tail pehle ke tip pe rakho. Bilkul shuru se bilkul ant tak ka arrow sum hai. Coordinates mein bas matching entries add karo: ( a , b ) + ( c , d ) = ( a + c , b + d )
Ab scaling aur adding combine karo:
Definition Linear combinationv 1 , … , v n linear combination koi bhi vector hai jo tum har ek ko scale karke aur results add karke build kar sakte ho:
c 1 v 1 + c 2 v 2 + ⋯ + c n v n . linear ka matlab hai hum sirf scale aur add karte hain — koi bending nahi, vectors ko aapas mein multiply nahi.
Intuition Yeh sab kuch ka dil kyun haiIndependence ki poori definition ek sentence hai linear combinations ke baare mein : "sirf woh linear combination jo origin pe land karti hai woh boring wali hai jahan har dial zero hai." Agar tum samajh gaye "har arrow scale karo, unhe add karo," tum pehle se machinery samajh gaye. Yeh Span and spanning sets se connect hota hai — span woh set hai saari linear combinations ka jo tum reach kar sakte ho.
Zero vector 0 length zero ka arrow hai — yeh origin pe shuru aur khatam hota hai, kahi point nahi karta. Iske coordinate list mein sab zeros hain: ( 0 , 0 ) ( 0 , 0 , 0 )
0 "kaun si combinations mujhe wahan wapas le jaati hain jahan main shuru hua tha?" Origin pe wapas land karna matlab arrows bilkul cancel ho gaye. Agar cancel karne ka sirf ek hi tarika hai woh kuch bhi na lena (c 1 = ⋯ = c n = 0 doosra tarika cancel karne ka ek redundant arrow expose karta hai.
0 0 Yeh sahi kyun lagta hai: dono almost identical dikhte hain.
Fix: 0 arrow hai — ek point. Plain 0 scalar hai. c 1 v 1 + ⋯ = 0
R n R n n saare number-lists ka collection hai. R 2 2 R 3 3 n dimension , available independent directions ki sankhya.
Intuition Is topic ke liye dimension kyun matter karta haiR n n n R 3 Basis and dimension dekho.
Ek matrix numbers ka rectangular grid hai. Is topic mein hum ek banate hain apne vectors ko columns ki tarah khada karke , side by side:
A = ∣ v 1 ∣ ∣ v 2 ∣ ⋯ ∣ v n ∣ . ∣
c c c = ( c 1 , c 2 , … , c n ) har column ka kitna lena hai.
A c = 0 c 1 v 1 + ⋯ + c n v n = 0 system of equations hai disguise mein. Ise A c = 0 homogeneous system hai (right side 0 null space banate hain.
Definition Trivial aur nontrivialc = 0 hamesha A c = 0 trivial solution hai.
Nontrivial solution koi bhi solution hai jahan kam se kam ek c i = 0
Situation
Matlab
Verdict
Sirf c = 0
Koi arrow redundant nahi
Independent
Koi c = 0
Ek arrow leftover combo hai
Dependent
Mnemonic Zero sirf boring tarike seIndependent = tum sirf boring tarike se 0 exciting tarika (ek dial on raha) matlab dependent.
Matrix ko row reduction se tidy karne ke baad (rows add/subtract karke leading entries ka staircase banao jise pivots kehte hain), rank pivots ki sankhya hai — genuinely independent columns ka count. Rank of a matrix dekho.
Bina pivot ke column ek free variable correspond karta hai — ek dial jise tum kuch bhi set kar sakte ho, jo turant ek nontrivial solution produce karta hai.
Intuition Pivots count karna poora sawaal kyun answer karta hai
Har column mein pivot (rank = n independent .Ek pivot missing (rank < n dependent .
Pivots count karna mechanical, guaranteed test hai — koi eyeballing nahi chahiye.
Determinant det A sirf tab defined jab A n R n signed area/volume measure karta hai. Determinant dekho.
Intuition Area independence kyun answer karta haiAgar R 2 zero area enclose karte hain — flat, squashed. Isliye:
det A = 0 det A = 0 A invertible hai.
det A Fix: det 3 × 4
Vector = arrow = number list
Scalar = stretch or flip number
Vector addition tip to tail
Linear combination scale and add
Vectors as columns matrix A
A times c equals combination
Homogeneous system Ac = 0
Trivial vs nontrivial solution
Khud ko test karo — sirf jawab dene ke baad reveal karo.
Main vector ( 3 , 2 ) x y ( 3 , 2 )
Main jaanta hoon kisi vector ko − 2 Length double ho jaati hai aur woh ulti direction mein flip ho jaata hai.
Main linear combination c 1 v 1 + c 2 v 2 v 1 c 1 v 2 c 2
Main 0 0 0 0
Main explain kar sakta hoon kyun A c A Har entry c i v i A c
Main jaanta hoon A c = 0 Sab-zero dial setting c = 0
Main jaanta hoon determinant use karne ki permission kab hai. Sirf square matrices ke liye — n R n
Main bata sakta hoon kyun free variable dependence force karta hai. Ek free dial nonzero set kiya ja sakta hai, jo A c = 0
Span and spanning sets — Section 3 ki saari linear combinations ka set.Basis and dimension — R n Rank of a matrix — Section 8 mein count kiye gaye pivots.Determinant — Section 9 ka area/volume test.Homogeneous systems and null space — equation A c = 0 Invertible matrix theorem — independent columns ⇔ invertible ⇔ det = 0 Linear independence — formal definition, testing — parent jiske liye yeh page tumhe taiyaar karta hai. 
